diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaors" "b/data_all_eng_slimpj/shuffled/split2/finalzzaors" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaors" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nHigh-resolution and high-cadence observations have shown the ubiquity of the magnetohydrodynamic (MHD) waves throughout the solar atmosphere \\citep[see, e.g.,][among many others]{Aschwanden99,Nakariakov99,Tomczyk07,Depontieu07,Mcintosh11,Morton15,Jafarzadeh17,Srivastava17}. MHD wave dissipation could play an important role in the heating of the solar corona \\citep[see, e.g.,][]{Hollweg78,Cranmer05,Cargill11,Mathioudakis13,Soler19,Tom20,Nakariakov20}, and the acceleration of the solar wind \\citep[see, e.g.,][]{Charbonneau95,Cranmer09,Matsumoto12,Shoda18}.\n\nTorsional Alfv\\'{e}n waves are a sub-type of axisymmetric Alfv\\'{e}n waves in cylindrical flux tubes. They are nearly incompressible, the restoring force is the magnetic tension, and their direction of polarization is perpendicular to the magnetic field lines. This kind of waves produce axisymmetric perturbations in the perpendicular components of velocity and magnetic field. Furthermore, unlike Alfv\\'{e}n waves with other azimuthal symmetries, torsional Alfv\\'{e}n waves do not couple with magneto-acoustic waves when the magnetic field is straight \\citep{Goossens11}. Torsional Alfv\\'{e}n waves have been reported in bright points \\citep{Jess09} and in a coronal active region structure \\citep{Kohutova20}. Torsional motions found in the chromosphere and transition region by \\citet{DePontieu12,DePontieu14} can also be interpreted as torsional Alfv\\'en waves. \\citet{Aschwanden20} interpreted oscillations in the magnetic energy during solar flares \nas torsional Alfv\\' {e}n waves. This kind of waves has also been reported in the solar wind during the interaction between coronal mass ejections \\citep{Raghav18}.\n\nIn coronal loops, closed magnetic structures with two footpoints anchored at the solar photosphere, there is no report of such waves. Nonetheless, torsional Alfv\\'{e}n waves can be excited at their footpoints through vortex\/twisting plasma motions in the solar photosphere \\citep[see, e.g.,][]{Fedun11,Shelyag11,Shelyag12,Wedmeyer12,Mumford15,Srivastava17b}. Recently, \\citet{Soler21} have investigated the propagation of torsional Alfv\\'{e}n waves from the photosphere to a coronal loop. As the waves reach the coronal loop, they can resonate with standing modes of the loop and drive global torsional oscillations \\citep[see also][]{Hollweg84,Hollweg84b}. \\citet{Soler21} found that large amounts of energy can be transmitted at coronal heights despite the chromospheric filtering. Particularly, they find that the energy flux is channeled mostly through the fundamental standing torsional mode of the loop.\n\nInspired by these results, in \\citeauthor{Diazsoler21} (\\citeyear{Diazsoler21}; hereafter Paper \\citetalias{Diazsoler21}), we investigated the nonlinear evolution of torsional Alfv\\'{e}n waves in a straight coronal flux tube with a constant axial magnetic field. In Paper \\citetalias{Diazsoler21}, see also the torsional model in \\citet{Guo19}, we showed that the phase mixing of the torsional Alfv\\'{e}n waves generates azimuthal shear flows. These flows eventually excite the Kelvin-Helmholtz instability (KHi), as \\citet{HeyvaertPriest83} and \\citet{Browning84} analytically predicted. Phase mixing is a linear phenomenon that occurs when waves in adjacent radial positions oscillate with different frequencies. The cause of phase-mixing is the presence of a frequency continuum, whose origin is in the inhomogeneities in density and\/or magnetic field \\citep[see, e.g.,][]{HeyvaertPriest83,Nocera84,Moortel00,Smith07,Prokopyszyn2019}. Phase-mixing increases the values of vorticity and current density and generates small scales, although at a relatively slow pace.\n\nThe KHi triggered by phase mixing can evolve nonlinearly forming large eddies that break into smaller eddies. This process initiates and drives turbulence. Turbulence accelerates the energy cascade to small scales, which might enhance the efficiency of wave energy dissipation \\citep{Hillier20}. There is evidence that coronal loops may be in a turbulent state \\citep{Demoortel14,Hahn14,Liu14,Xie17}. The generation of turbulence is also a result obtained by numerous studies of numerical simulations of kink oscillations of coronal loops \\citep[see, e.g.,][among others]{Terradas08,Terradas18,Antolin15,Magyar16,Howson17a,Karampelas19,Antolin19}. \n\n\n\nIn the present paper, we extend the investigation of Paper~\\citetalias{Diazsoler21} by replacing the constant axial magnetic field by a twisted magnetic field. The existence of twisted coronal loops has been confirmed by observations \\citep[see, e.g.,][]{Kwon08,Aschwanden12,Thalmann14,Aschwanden19}. The behavior of linear MHD waves in twisted flux tubes has extensively been investigated analytically or semi-analytically \\citep[see, e.g.,][among many others]{Bennet99,Erdelyi06,Erdelyi07,Erdelyi10,Rudermann07,Karami09,Karami10,Terradas12,Ebrahimi17,Ebrahimi22}. The nonlinear evolution of kink MHD waves in twisted flux tubes has also been investigated numerically \\citep{Howson17a,Terradas18}. Here, we explore how the presence of magnetic twist affects the nonlinear evolution of torsional Alfv\\'en waves.\n\n\\section{Numerical setup}\n\\label{Sect_Num}\n\\subsection{Initial Configuration} \\label{Subsect_model}\n\nAs in Paper~\\citetalias{Diazsoler21}, we use the standard coronal loop model \\citep[see, e.g.,][]{Edwinroberts83}. The model is made of an overdense cylindrical flux tube of radius, $ R $, and length, $ L, $ embedded in a low-$\\beta$ uniform coronal environment. We set $L\/R = 10$. We used a shorter loop length than what is reported from observations, typically $L\/R \\sim 100$. The main reason for doing so is to speed up the simulation times. The periods of the standing torsional oscillations are proportional to the loop length. Thus, if a longer loop were considered, the periods and simulation times would be longer, but the dynamics would be essentially the same. The effect of considering larger loop lengths was explored in Paper~\\citetalias{Diazsoler21}.\n\n\nThe loop footpoints are fixed at two rigid walls representing the solar photosphere. We neglect the curvature of the coronal loop and the thin chromospheric layer at the loop feet. In Paper~\\citetalias{Diazsoler21} we assumed a uniform magnetic field along the cylinder axis. Here, we improve the model by considering a twisted magnetic field, which is a more realistic representation of the magnetic field in coronal loops, namely\n\\begin{equation}\n {\\bf B} = B_{\\varphi}(r) \\hat{e}_\\varphi + B_{z}(r) \\hat{e}_z, \n\\end{equation}\nwhere $B_{\\varphi}$ and $B_{z}$ are the azimuthal and longitudinal components of the background magnetic field in a cylindrical coordinate system denoted by $r$, $\\varphi$, and $z$. In our model, the $z$-axis coincides with the flux tube axis. We consider the same force-free twist model as that used in \\citet{Terradas18}, namely\n\\begin{eqnarray}\nB_{\\varphi}(r)&=&\\left\\lbrace \\begin{matrix} cr\/R, & \\mbox{if}\\;r < R,\\label{bphi} \\\\\n cR\/r, & \\mbox{if}\\;r\\geq R,\\end{matrix}\\right. \\\\\nB_{z}(r)&=&\\left\\lbrace \\begin{matrix} \\sqrt{B^{2}_{0}+2c^{2}\\left(1-r^{2}\/R^{2}\\right)}, & \\mbox{if}\\;r < R, \\label{bz}\\\\\n B_{0}, & \\mbox{if}\\;r\\geq R,\\end{matrix}\\right. \n\\end{eqnarray}\nwhere $ B_{0} $ is a constant that corresponds to the magnetic field strength at the tube axis and $c$ is a dimensionless parameter that controls the amount of magnetic twist. \n\nThe largest value of the magnetic twist parameter used here is $ c = 0.4$, which is chosen according to observational constraints. \\citet{Aschwanden13p3} computed the twist angle, $ \\theta $, in coronal loops as $ \\theta=\\arctan \\sqrt{q_{\\mathrm{free}}}$, where $ q_{\\mathrm{free}}$ is the free energy ratio. The calculation of $q_{\\mathrm{free}}$ can be achieved from observations through line-of-sight magnetograms, the three-dimensional (3D) position of the coronal loop, and a nonlinear force-free field code. Tables 3 in \\citet{Aschwanden14} and \\citet{Aschwanden16} show that the free energy ratio ranges from 0 to $\\sim 0.25 $. Therefore, twist angles lower than $ \\sim 25^{\\circ}$ are expected. The twist angle of our model can be calculated as\n\\begin{equation}\n\\theta=\\arcsin\\left(\\frac{B_{\\varphi}(r)}{B(r)}\\right),\n\\label{angles}\n\\end{equation}\nwhere $ B(r)=\\sqrt{B^{2}_{\\varphi}\\;(r)+B^{2}_{z}\\;(r)} $ is the modulus of the magnetic field.\nFor $c=0.4$, the twist angle is $ \\theta =21.8^{\\circ}$ at $r=R$, which is slightly lower than the largest angle inferred by \\citet{Aschwanden14} and \\citet{Aschwanden16}. On the other hand, the number of turns of the magnetic field lines over the cylinder length can be computed as,\n\\begin{equation}\nN\\left(r\\right)=\\frac{1}{2\\pi} \\frac{L}{r} \\frac{B_{\\varphi}\\left(r\\right)}{B_{z}\\left(r\\right)},\n\\label{ntw}\n\\end{equation}\nwhere the factor $L B_{\\varphi}\\left(r\\right)\/rB_{z}\\left(r\\right) \\equiv \\Phi(r) $ is the absolute twist. \\citet{Kwon08} considered a sample of 14 loops from Transition Region And Coronal Explorer (TRACE; \\citeauthor{TRACE} \\citeyear{TRACE}) observations in extreme ultraviolet and determined that their absolute twist values ranged from $0.22\\pi$ to $1.73\\pi$. This result implies that $0.11 < N < 0.865$. At $ r=R $ in our model, where $B_{\\varphi} $ is maximum, we obtain $ N= 0.64$ for $c=0.4$ and $ L\/R= 10$, so that the maximum twist considered here is below the upper limit of the interval inferred by \\citet{Kwon08}.\n\n\n\nA twisted magnetic field is prone to develop the kink instability. In the case of uniform twist, \\citet{HoodPriest79} found that the kink instability can develop when the absolute twist is larger than $ 3.3\\pi $. With an improved analysis, \\citet{HoodPriest81} reduced the threshold value to $ 2.49\\pi $. Using the definition of $N$ in Eq.~(\\ref{ntw}), we find that the kink instability may appear in our model if $ N \\geq 1.65$ according to the critical twist of \\citet{HoodPriest79} and if $ N \\geq 1.245 $ according to the critical twist of \\citet{HoodPriest81}. Clearly, we are sufficiently far away from the critical twist for the kink instability because our maximum twist ($ N= 0.64$) is still relatively weak.\n\nWe note that we used a pre-existing twisted magnetic field in our model. A different way to proceed would be to create twist in the model by dynamically rotating the magnetic field lines anchored in the photosphere, as done in \\cite{Ofman09}. This alternative approach imposes a more complex scenario and it is not considered here for simplicity.\n\nSince the magnetic field is force-free, we consider in the model a background uniform gas pressure, $p_0$. In turn, the equilibrium density, $\\rho_0$, is the same as that used in Paper \\citetalias{Diazsoler21}. Thus, we used\n\\begin{equation}\n\\rho_{0}(r)=\\left\\{\n\\begin{array}{lll}\n\\rho_{\\mathrm{i}}, & \\mbox{if} & r \\leq R-\\frac{l}{2},\\\\\n\\rho_{\\mathrm{tr}}(r), & \\mbox{if} & R-\\frac{l}{2} 10^3 $ in our analysis owing to the possible role of numerical dissipation at the smallest scales.\n\n\\section{Concluding remarks}\n\\label{Sect_con}\n\nWe conclude the investigation started in Paper~\\citetalias{Diazsoler21} about the ability of torsional Alfv\\'en waves to drive turbulence in coronal loops. Because of plasma and\/or magnetic nonuniformity, standing torsional Alfv\\'{e}n waves undergo the process of phase mixing, generating shear flows perpendicular to the magnetic field direction in adjacent radial positions as time increases. Eventually, the shear flows trigger the KHi, as \\citet{HeyvaertPriest83} and \\citet{Browning84} first predicted.\n\nIn the absence of magnetic twist (the case studied in Paper~\\citetalias{Diazsoler21}), the KHi can grow nonlinearly without opposition. The nonlinear evolution of the KHi naturally induces turbulence. First, large KHi eddies are formed, which later break into smaller eddies in a cascade-like dynamics. Mixing of plasma takes place. An important increase in vorticity is found and the generation of small scales speeds up compared with the initial phase dominated by phase mixing alone. Turbulence evolves perpendicularly to the magnetic field, so that for a straight magnetic field turbulence is pseudo-2D.\n\nIn the present paper, we have explored the role of magnetic twist. While the linear evolution of phase mixing is similar in the presence or in the absence of twist, the dynamics of the subsequent KHi growth and turbulence generation depends strongly on the strength of the twist.\n\nIf magnetic twist is sufficiently weak, the onset and growth of the KHi is just delayed, but the dynamics is similar to that in the case of a straight magnetic field. In this regime, the stronger the magnetic twist, the longer the delay. Although the vorticity still increases dramatically during the development of turbulence, the increase is smaller compared with that in absence of magnetic twist. Small scales are still quickly generated by turbulence, although at a slower pace than when twist is absent. Turbulence still evolves perpendicularly to the magnetic field lines, but since the field is twisted, turbulence is no longer confined to perpendicular planes to the tube axis. \n\nConversely, under the presence of a strong enough magnetic twist, the scenario is completely different. If the strength of magnetic twist surpasses a critical value, or a critical twist angle, the KHi vortices cannot grow. The KHi itself is still locally excited by the phase-mixing generated shear flows, but the tension of the twisted magnetic field prevents now the nonlinear development of the instability \\citep[see, e.g.,][]{Galinsky94,Ryu2000,Hillier19,DiazSuarezsolerlet2021}. As a consequence, there is no enhancement of vorticity, neither turbulence is driven. Thus, the generation of small scales is not accelerated and continues slowly evolving at the rate dictated by phase-mixing. \n\nAlthough the effect of magnetic twist may oppose the process, the nonlinear evolution of torsional Alfv\\'en waves remains as a viable mechanism to induce turbulence as long as coronal loops are weakly twisted. There are other effects that should be explored in the future. For instance, the tension of the background field in curved coronal loops may also affect the triggering and evolution of the KHi and associated turbulence. It could be also interesting to investigate turbulence generation in multithreaded loops \\citep[see, e.g.,][]{Ofman09}.\n\n\n\\begin{acknowledgement}\nThis publication is part of the R+D+i project PID2020-112791GB-I00, financed by MCIN\/AEI\/10.13039\/501100011033. S.D. acknowledges the financial support from MCIN\/AEI\/10.13039\/501100011033 and European Social Funds for the predoctoral FPI fellowship PRE2018-084223. We also thank the UIB for the use of the Foner cluster. For the simulation data analysis we have used VisIT \\citep{VisIt} and Python 3.6. In particular, we have used Matplotlib \\citep{Hunter07}, Scipy \\citep{Virtanen20}, and Numpy \\citep{Harris20}. We are thankful to B. Vaidya and his contributors for the tool pyPLUTO.\n\\end{acknowledgement}\n\n\n\\bibpunct{(}{)}{;}{a}{}{,} \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\nX-ray Binaries (hereafter XrBs) display cycles of strong activity, where their luminosity increases by several orders of magnitude and their spectral shape changes drastically on long timescales, before decreasing back to quiescence. We call this entire phase an outburst. At the beginning of such an outburst, they are in the so-called hard state where their X-ray spectra are dominated by a power-law component with a hard photon index $\\Gamma<2$ \\citep{Remillard06}. During this state, they also show flat or slightly inverted radio spectra \\citep[e.g.,][]{CorbelFender02}, interpreted as self-absorbed synchrotron emission from collimated, mildly relativistic jets \\citep{BK79}. At some point, the X-ray spectrum of these objects undergoes a smooth transition from this power-law dominated spectral shape, to a dominant blackbody of temperature $\\sim~1$~keV only. In addition, steady radio emission disappears, suggesting a quenching of the jets \\citep{Corbel04, Fender04, Fender09}. X-ray binaries remain in this soft state until a decline in luminosity makes them transit back to a hard state at the end of the outburst, along with reappearance of the jets. This surprising behavior has been observed multiple times in the past decades, and in dozens of different objects, where some have even undergone multiple outbursts \\citep[see][for a global overview]{Dunn10}. What is even more striking in these outbursts, is that they seem to be very similar in different objects, while being different from one outburst to another in the same object! \\\\\n\n\nA general scenario has been proposed by \\citet{Esin97}, where changes in the accretion flow geometry provokes the spectral variations. In this view, the interplay between two different accretion flows is responsible for the spectral changes in the disk: in the outer parts, a cold standard accretion disk \\citep[SAD, ][]{SS73} extends down to a given truncation (or transition) radius where an advection dominated hot flow\\footnote{The inner hot flow is often referred to as a \"hot corona\". However, this designation remains ambiguous and we choose not to utilize it.} takes place \\citep{Ichimaru77, Rees82}. The inner hot flow is expected to be responsible for the power-law component, while the outer cold flow produces the blackbody radiation. While the presence of a SAD in the outer regions of the disk is highly accepted to date \\citep{Done07}, the physical properties of the advection dominated inner flow remain an open question.\n\nBetween Slim disks \\citep{Abramowicz88}, ADAFs \\citep{Narayan94}, ADIOS \\citep{BB99}, LHAFs \\citep{Yuan01} or more peculiar models \\citep[e.g.,][]{Meyer00, Lasota01}, no satisfactory explanation has been provided so far \\citep{Yuan14}. A discussion about the major models and their current state can be found in \\citet{paperII}. Many questions remain open but in this article we focus on: (1) reproducing the X-ray spectral shape of all the generic spectral states, (2) explaining the correlated accretion-ejection processes through their observables, i.e. radio and X-rays fluxes. \\\\\n\n\n\nIn this work we consider an accretion flow extended down to the inner-most stable circular orbit, and thread by a large scale vertical magnetic field $B_z$.\nIt is well known that matter can only accrete by transferring away its angular momentum. This can be achieved by few physical mechanisms, namely internal turbulent (\"viscous\") torques and magnetic torques from an outflow:\n\\begin{itemize}\n\\item When accretion is mostly due to internal (turbulent) viscosity, angular momentum is transported radially. This produces an optically thick and geometrically thin accretion disk, a SAD, which is observed as a cold multicolor-disk blackbody. The production of winds by SAD is still a debated question, but few simulations and observations have shown the possible existence of winds from standard disks (see discussion in section~\\ref{sec:innerSAD}). However, these winds cannot explain the powerful jets associated with XrBs so that the SAD suits perfectly for the jet-less thermal states in XrBs, i.e. soft states.\n\\item Alternatively, self-confined super-Alfv\\'enic jets can also provide a feedback torque on the disk, carrying away vertically both energy and angular momentum. This accretion mode, referred to as jet-emitting disks \\citep[JED, ][and subsequent work]{Ferreira95}, presents a supersonic accretion speed. Therefore, for the same accretion rate, it has a much smaller density than the SAD, leading to optically thin and geometrically thick disks. Disks accreting under this JED mode are thus good candidates to explain power-law dominated and jetted states in XrBs, i.e. hard states.\n\\end{itemize}\n\n\\noindent The magnetic field strength is characterized by the mid-plane magnetization $\\mu(r)= B_z^2\/\\mu_o P_{tot}$, where $P_{tot}$ is the total pressure, the sum of the kinetic plasma pressure and the radiation pressure. At large magnetization, the SAD can no longer be maintained as magneto-centrifugally driven jets are launched: a JED arises \\citep{Ferreira95,Ferreira97}. Full MHD calculations of JEDs have shown that the transition occurs around $\\mu \\sim 0.1$ \\citep{Casse00a,Casse00b,Lesur13}. \\\\\n\n\n\n\nA global scenario based on these possible dynamical transitions in accretion modes has then been proposed to explain XrB cycles (\\citealt{Ferreira06}, hereafter paper I; \\citealt{Petrucci08}). What would be the causes of the evolution of the disk magnetization distribution $\\mu(r)$ is still a highly debated question. The main uncertainty comes from the interplay between the magnetic field advection and diffusion in turbulent accretion disks, geometrically thin or thick and with or without jets. Modern global 3D MHD simulations do show that large scale magnetic fields are indeed advected \\citep{Avara16, ZhuStone18}, but these simulations are always done on quite short time scales, up to few seconds, and it is hard to scale them to the duration of XrB cycles, typically lasting over several month. In this paper, we assume that cycles result from transitions in accretion modes and focus on their observational consequences. \\\\\n\n\n\\citet{Petrucci10} computed the thermal states of a pure JED solutions and successfully reproduced the spectral emission, jet power and jet velocity during hard states of Cygnus X-1. However, their calculations were done assuming a one temperature (1T) plasma, but the necessity of a (2T) plasma seem inevitable to cover the large variation of accretion rate expected during an entire outburst \\citep{Yuan14}.\n\n\\citet{paperII}, hereafter paper II, extended this work by developing a two-temperature (2T) plasma code that computes the disk local thermal equilibrium, including advection of energy and addressing optically thin-to-thick transitions in both radiation and gas supported regimes.\nFor a range in radius and accretion rates, they showed that JEDs exhibit three thermal equilibria, one thermally unstable and two stable ones. Only the stable equilibria are of physical importance \\citep{Frank92}. One solution consists of a cold plasma, leading to an optically thick and geometrically thin disk, whereas the second solution describes a hot plasma, leading to an optically thin and geometrically thick disk. Due to the existence of these two thermally stable solutions a hysteresis cycle is naturally obtained. But large outbursting cycles, such as those exhibited by GX 339-4, cannot be reproduced (paper II). Nevertheless, JEDs have the striking property of being able to reproduce very well hard states spectral shape, all the way up to very luminous hard states $L > 30 \\% L_{Edd}$.\\\\\n\n\nHowever, SAD-JED local transitions are expected to occur locally on dynamical time scales, typically $\\sim 1$~ms Kepler orbital time at $10~R_g$, whereas hard-to-soft transitions involve time scales of days or even weeks. This implies that, at any given time, the disk must be in some hybrid configuration with some regions emitting jets, while others do not. It is expected that jets, namely magnetocentrifugally-driven flows \\citep{BP82}, are only launched from the disk innermost regions. This translates into a hybrid configuration where an inner JED is established from the last stable orbit $R_{in}$ until an unknown transition radius $R_J$, and then surrounded by an outer SAD until $R_{out}$. The exact location of the transition $R_J$ depends on the global response of the magnetic field $B_z$ to accretion rate evolution at the outer edge $\\dot{M}_{out}$, two unknowns. Hence $R_J$ is treated here as a free parameter of the model. Such radial transition between two flows has already been studied in the context of non-magnetized accretion flows, advocating for mechanism such as evaporation or turbulent diffusion at the origin of the transition \\citep[see, e.g.,][]{Meyer94, Honma96}. We study however configurations where the magnetization $\\mu$ is large (near equipartition) and uniform in the JED region, and drops at the transition radius $R_J$, organizing the two-flow structure. Although the main properties of isolated JED and SAD are well understood, hybrid configurations imply mutual interactions that need to be described. For instance, part of cold radiation emitted from the SAD region must be intercepted by the geometrically thick JED and provides an additional cooling term that might change its general properties. \\\\\n\nIn this paper, we explore the observational signatures of disk configurations with an inner JED and an outer SAD. Section~\\ref{sec2} describes this hybrid configuration, including interactions between the two regions, and explores some of its dynamical consequences. Section~\\ref{sec3} presents the procedure followed to simulate and fit synthetic X-ray data from our theoretical spectra as well as to estimate the jet radio emission. Section~\\ref{sec4} is devoted to the exploration of the parameter space, by varying the disk accretion rate $\\dot{M}_{in}$ and transition radius $R_J$. Playing with these two parameters allows to completely cover the disk fraction luminosity diagram \\citep[hereafter DFLD, ][]{Kording06}. As an illustrative example, we apply our model and reproduce canonical states of GX 339-4, both in X-ray spectral shape and radio fluxes. We end with concluding remarks in section~\\ref{sec:Ccl}.\n\n\n\\section{Hybrid disk configuration: internal JED and external SAD} \\label{sec2}\n\n\\subsection{General properties} \n\\label{sec:GeneralPpts}\n\nAs introduced in paper II, we consider an axisymmetric accretion disk orbiting a black hole of mass $M$. For simplicity, the disk is assumed to be in global steady-state so that any radial variation of the disk accretion rate $\\dot{M}(R)$ is only due to mass loss in outflows. We define $H(R)$ the half-height of the disk, $\\varepsilon(R) = H\/R$ its aspect ratio, $\\dot{M}(R) = - 4 \\pi R u_R \\Sigma$ the local disk accretion rate, $u_R$ the radial (accretion) velocity and $\\Sigma = \\rho_0 H$ the vertical column density with $\\rho_0$ the mid plane density. Throughout the paper, calculations are done within the Newtonian approximation. Moreover, and for the sake of simplicity, the disk is assumed to be always quasi-Keplerian with a local angular velocity $\\Omega \\simeq \\Omega_K = \\sqrt[]{G M R^{-3}}$, where $G$ is the gravitational constant. \\\\\n\nThe disk is assumed to be thread by a large scale vertical magnetic field $B_z(R)$. We assume that such a field is the result of field advection and diffusion and we neglect thereby any field amplification by dynamo.\nClearly, the existence of cycles shows that some evolution is ongoing within the disk. However, the timescales involved (days to months) are always much longer than accretion timescales inferred from the X-ray emitting regions. Thus, as for any other disk quantity, the local magnetic field is assumed to be stationary on dynamical time scales (Keplerian orbital time). \\\\\n \n\n\\begin{figure}[h!\n \\centering\n \\includegraphics[width=\\columnwidth]{config.pdf} \n \\caption{Example of hybrid disk configuration in the JED-SAD paradigm. The inner disk regions are in a jet-emitting disk (JED) mode, up to a transition radius $r_J$, beyond which a standard accretion disk (SAD) is settled. The disk scale height $H(R)$ is accurately displayed, while colors corresponds to the central electronic temperature $T_e$ in Kelvin. The disk switches from an outer optically thick, geometrically thin jet-less disk to an inner optically thin, geometrically thick disk launching self-confined jets (not shown here). This solution has been computed for a transition radius $r_J=15$ and a disk accretion rate $\\dot{m}_{in}=0.1$ at the disk inner radius $r_{in}=2$ (see section~\\ref{sec:GeneralPpts} for more details). Other similar examples are shown in Fig.~\\ref{fig:JoliesImages} for different pairs $(\\dot{m}_{in}, r_J)$.}\n \\label{fig:config}\n\\end{figure\n\nThe hybrid disk configuration is composed of a black hole of mass $M$, an inner jet-emitting disk from the last stable orbit $R_{in}$ to the transition radius $R_{J}$ and an outer standard accretion disk from $R_{J}$ to $R_{out}$. The system is assumed to be at a distance $D$ from the observer. In the following, we adopt the dimensionless scalings: $r = R\/R_g$, $h = H\/R_g= \\varepsilon r$, where $R_g = GM\/c^2$ is the gravitational radius, $m = M\/M_{\\odot}$, and $\\dot{m} = \\dot{M}\/\\dot{M}_{Edd}$, where $\\dot{M}_{Edd} = L_{Edd} \/ c^2$ is the Eddington accretion rate\\footnote{Note that this definition does not include the accretion efficiency, usually of the order $\\sim 10\\%$ for a Schwarzschild black hole. This means that reaching Eddington luminosities would require $\\dot{m}_{in} \\gtrsim 10$ (see Fig.~\\ref{fig:Lxmdot}).} and $L_{Edd}$ is the Eddington luminosity. Since GX 339-4 appears to be an archetypal object, we decided to concentrate only on this object. We thus use a black hole mass $m = 5.8$, a spin $a = 0.93$ corresponding to $r_{in} = 2.1$ and a distance $D = 8$~kpc \\citep{Miller04}\\footnote{See also \\citet{Munoz08}, \\citet{Parker16} or \\citet{Heida17} for more recent estimations.}. All luminosities and powers are expressed in terms of the Eddington luminosity $L_{Edd}$. An example of disk configuration is shown in Fig.~\\ref{fig:config} for $\\dot{m}_{in} = \\dot{M}_{in}\/\\dot{M}_{Edd} = 0.1$ and $r_J = R_J \/ R_g = 15$.\\\\\n\nOur goal is then to compute, as accurately as possible, the radial disk thermal equilibrium from $r_{out}$ to $r_{in}$ by taking into account the known dynamical properties associated with each accretion mode. The inflow-outflow sctructure is described by the following midplane quantities:\n\\begin{eqnarray}\n\\mu &= &\\frac{B_z^2\/\\mu_0}{P_{tot}} = \\frac{B_z^2 \/ \\mu_0}{P_{gas} + P_{rad}} \\nonumber \\\\\n\\xi &=& \\frac{d \\ln \\dot{m}}{d \\ln r} \\nonumber \\\\\nm_s &=& \\frac{-u_R}{c_s} = \\frac{-u_R}{\\Omega_K H} =m_{s,turb} + m_{s, jet} = \\alpha_{\\nu} \\varepsilon + 2 q \\mu \\\\\n b &= & \\frac{2 P_{jet}}{P_{acc, JED}} \\nonumber\n\\end{eqnarray}\nwhere $\\mu$ is the disk magnetization, $\\xi$ the local ejection index, $m_s$ the sonic accretion Mach number and $b$ the fraction of the JED accretion power $\\displaystyle P_{acc, JED} = \\left [ \\frac{G M \\dot{M}}{2 R} \\right ]^{R_{in}}_{R_{J}}$ that is carried away by the jets in the JED. The parameter $q \\simeq -B_\\phi^+\/B_z $ is the magnetic shear of the magnetic configuration and $B_\\phi^+$ is the toroidal magnetic field at the disk surface \\citep[see][for more details]{Ferreira97}. From the above radial distributions, we can deduce the expressions of the vertical magnetic field $B_z$, the accretion speed $u_R$ and the disk surface density $\\Sigma$ as a function of $\\dot{m}_{in}$ and the disk aspect ratio $\\varepsilon=H\/R$. The latter is obtained by solving the coupled energy equations for the ions and electrons in order to compute both electronic and ion temperatures (see paper II for a full description of the method). \n\n\n\\subsubsection{Inner JED}\n\nJet-emitting disks solutions from a large radial disk extent are known to exist in a restricted region of the parameter space \\citep[][Fig. 2]{Ferreira97}. For simplicity, we assume that any given configuration is stationary and that parameters are constant. An extensive study of the thermal structure and associated spectra of JED can be found in paper II, where it is shown that the following set of parameters best reproduces XrB hard states, from low to very high luminosities: \n\\begin{itemize}\n\\item $\\mu= 0.5$: the disk magnetization $\\mu$ has very little influence on X-ray spectra, because the synchrotron emission does not contribute much to the equilibrium (paper II). This value has thus been chosen to lie between the two extreme values allowed for JED solutions, namely $\\mu_{min}=0.1$ and $\\mu_{max}=0.8$ \\citep{Ferreira97}.\n\\item $\\xi=0.01$: the smaller the ejection index, the less mass is being ejected and the larger the asymptotic jet velocity. A value $\\xi=0.01$ is consistent with mildly relativistic speeds \\citep[e.g., the case study for Cyg X-1 in][]{Petrucci10}.\n\\item $m_s=1.5$: within the JED accretion mode, the jets torque is dominant and imposes $m_s= m_{s, jet} =2 q \\mu $. The precise value of $m_{s,jet}$ depends on the trans-Alfv\\'enic constraint, but accretion in a JED is always at least sonic and usually supersonic $m_{s,jet}>1$ \\citep{Ferreira97}. In paper II we showed that a supersonic accretion with $m_s=1.5$ allows to reproduce luminous hard states. \n\\item $b=0.3$: the fraction of the released accretion energy $P_{acc}$ transferred to the jets has been computed within self-similar models and goes from almost $1$ to roughly $0.2$ \\citep{Ferreira97, Petrucci10}. The chosen value also appears as a good compromise and facilitates the reproduction of luminous hard states (paper II).\n\\end{itemize}\n\n\n\n\\noindent The fact that the disk accretion rate necessarily varies with the radius has been first introduced in accretion-ejection models by \\citet{Ferreira93a}, in the context of magnetically driven jets, and later by \\citet{BB99} in the context of thermally-driven outflows. In both cases, the disk ejection efficiency is characterized by the radial exponent $\\xi$ in $\\dot m (r) \\propto r^\\xi$. While it has been shown that magnetically-driven jets require $\\xi < 0.1$ \\citep{Ferreira97}, the values measured in many simulations is usually higher, lying between $0.5$ and $1$ \\citep[][and references therein]{CasseKeppens04, YBW12, Yuan15}. It is somewhat troublesome that different simulations lead to a comparable value regardless of the strength of the magnetic field. Moreover, they were mostly done in the context of non-radiating hot accretion flows. But on the other hand, \\citet{ZhuStone18} obtained $\\xi \\sim 0.003$ with an isothermal equation of state. Our guess is that this issue is not settled yet, especially given the extreme sensitivity of the disk ejection efficiency to the local thermodynamics \\citep{Casse00b}. As discussed above, we thus simply assume a small value for $\\xi$ that is compatible with the existence of relativistic jets.\n\n\n\n\n\n\n\n\\subsubsection{Outer SAD} \\label{sec:innerSAD}\n\nAs argued in introduction, the outer disk regions are assumed to accrete under the SAD mode. This implies that the relevant torque is turbulent, probably due to the magneto-rotational instability (MRI hereafter). In this case, $m_{s,turb} = \\alpha_{\\nu} \\varepsilon $. The Shakura-Sunyaev viscosity parameter $\\alpha_{\\nu}$ needs to be specified and we use $\\alpha_{\\nu}=0.1$ throughout this paper \\citep{Hawley02, King04, Penna13}. The magnetization must be small enough to allow for the development of the MRI, we choose $\\mu=10^{-3}$. As long as the SAD remains optically thick the value of $\\mu$ does not affect our calculations of the SAD thermal equilibria. \\\\\n\n\nBy definition, no jets are present in a SAD. This translates into $\\xi_{SAD}=0$ (no mass loss) and $b_{SAD}=0$ so that all released energy is either radiated or advected by the flow. Doing so, we neglect the potential presence of winds usually observed in XrB, especially at high luminosities \\citep{Ponti12, Tetarenko16}. This assumption sounds reasonable for two reasons. First, although mass loss from turbulent disks is indeed possible and actually observed in MHD simulations \\citep[e.g.,][]{Proga00, BaiStone13, Suzuki14, Bethune17, ZhuStone18}, these magneto-thermally driven flows carry away a negligible fraction of the disk angular momentum and released accretion energy, introducing thereby no significant change in the disk structure. Second, the mass rate feeding the inner JED is simply $\\dot{m}(r_J)= \\dot{m}_{in} (r_J\/r_{in})^{\\xi}$ and is independent from $\\xi_{SAD}$. Increasing $\\xi_{SAD}$ up to say, $0.5$, would imply a strong increase of the disk accretion rate in the regions beyong $r_J$ up to $r_{out}$. This would of course lead to a significant change of the emitted spectrum from these outer regions, but with no detectable counterpart \\citep[see however][and references therein]{Susmita16} as long as the disk remains optically thick, which is the case here.\n\n\n\n\n\n\\subsubsection{JED-SAD radial transition}\n\\label{sec:inter}\n\nWe examine here some properties of the transition, assuming that it occurs over a radial extent of the order of a few local disk scale heights. \\\\\n\nThe first striking property is the existence of a trans-sonic critical point near $R_J$. Indeed, while the accretion flow is subsonic in the SAD with a Mach number $m_{s,SAD} = m_{s,turb} = \\alpha_{\\nu} \\varepsilon_{SAD} \\ll 1$, it is supersonic in the inner JED with $m_{s,JED} = m_s = 1.5$. This property is a natural consequence of the transition from a turbulent \"viscous\" torque acting within the outer SAD to a dominant jet torque in the JED. Since the disk is assumed to be in a steady-state, the continuity of the mass flux $\\dot{M}_{JED}=\\dot{M}_{SAD}$ must be fulfilled at the transition radius. Given the difference between the JED and SAD sonic Mach number, this implies a drastic density decrease $\\displaystyle \\Sigma_{JED}\/\\Sigma_{SAD} \\sim \\alpha_{\\nu} \\varepsilon_{SAD}^2\/\\varepsilon_{JED} \\ll 1$ between the SAD and the JED. The Thomson optical depth being defined by $\\tau_T \\propto \\Sigma$, this density drop therefore implies a huge drop in the disk optical depth. Thus, a dynamical JED-SAD radial transition naturally goes with an optically thin - optically thick transition (see paper II). \n \nThe second striking property is the possible existence of a thin super-Keplerian layer between the JED and the SAD. In the outer SAD, the disk is slightly sub-Keplerian with a deviation due to the radial pressure gradient and of the order of $\\varepsilon_{SAD}^2$. Within the JED, the much larger magnetic radial tension leads to a larger deviation of the order of $\\mu \\varepsilon$ \\citep{Ferreira95}. This requires that the radial profile $\\Omega(R)$ has two extrema (with $d\\Omega\/dR \\simeq 0$). Since all the disk angular momentum is carried away vertically in JEDs, there is no outward angular momentum flux into the SAD at $R_J$. This translates into a \"no-torque\" condition for the SAD. Such a situation has already been discussed in the context of a radial transition between an outer cold (SAD) and inner hot (ADAF) accretion flows, leading to a super-Keplerian layer \\citep{Honma96, Abramowicz98}. It is, however, not clear whether such a thin layer would still be present in our context given the existence of magnetic forces. But these two extrema of the angular velocity clearly define the radial end points of each dynamical (SAD or JED) solution, and the trans-sonic transition occurring in-between. \\\\\n\n\nConstructing a dynamical solution describing the radial transition between a SAD and a JED is beyond the scope of the present paper and will be studied elsewhere.\nFrom now on, we assume that the two accretion modes can always be matched at a transition radius $R_J$. The calculation of the global disk equilibrium can then be undergone using $\\dot{m}_{in} = \\dot{M}_{in} \/ \\dot{M}_{Edd}$ and $r_J = R_J \/ R_g$ as independent variables.\n\n\n\n\n\n\n\\subsection{Thermal structure of hybrid JED-SAD configurations}\n\\label{sec:discussionJED}\n\nAs described in paper II, our accretion flow is locally described by a two-temperatures ($T_e$, $T_i$), fully ionized plasma of densities $n_e = n_i$, embedded in a magnetic field $B_z$. We recall here the main equations used to compute the thermal equilibrium (see paper II for full explanations). The electron and proton temperatures are computed at each radius using the coupled steady-state local energy balance equations\n \\begin{eqnarray}\n \\left( 1-\\delta \\right) \\cdot q_{turb} &=& q_{adv,i} + q_{ie} ~~~~~~~~~~~~~~~~~~~~ \\text{ions} \\label{eq:IONS} \\\\\n \\delta \\cdot q_{turb} &=& q_{adv,e} - q_{ie} + q_{rad} ~~~~~~~~~ \\text{electrons} \\label{eq:ELECTRONS}\n \\end{eqnarray}\nwhere the local heating term, of turbulent origin, varies according to the radial zone considered. Within the JED in $RR_J$ it is \n\\begin{equation}\nq_{turb,SAD} = \\frac{3 G M \\dot{M}(R)}{8 \\pi H R^3} \\left( 1 - \\sqrt[]{R_{J}\/R} \\right) \\label{eq:qturbSAD}\n\\end{equation}\nmaking use of the no-torque condition imposed at $R_{J}$ and a constant $\\dot{M}$ at the transition (section~\\ref{sec:inter}). In principle, the released turbulent energy could be unevenly shared between ions and electrons by a factor $\\delta$. Throughout this paper, we use $\\delta=0.5$ \\citep[see paper II,][section 2.3 and references therein]{Yuan14}. The other terms appearing in Eq.~(\\ref{eq:IONS}) and (\\ref{eq:ELECTRONS}) are the ion (electron) advection of internal energy $q_{adv,i}$ ($q_{adv,e}$), the Coulomb collisional interaction between ions and electrons $q_{ie}$ and the radiative cooling term due to the electrons $q_{rad}$. This term, as well as the radiation pressure $P_{rad}$ term, is computed using a bridge function allowing to accurately deal with both the optically thin and the optically thick regimes (paper II). The optically thin cooling regime $q_{thin}$ is computed with the BELM code \\citep{Belmont08}, which includes Compton scattering, emission and absorption through bremsstrahlung and synchrotron processes.\n\n\nThe thermal equilibrium of the SAD region is well known. For large accretion rates, required for outbursting XrBs, the disk is mostly in the optically thick, geometrically thin cold regime ($T_e \\sim 10^5-10^7$~K). \nHeating of the SAD surface layers by the hard X-rays emitted by the inner JED might produce some disk evaporation \\citep[][]{Meyer94, Meyer00, Liu05, Meyer05}. This would require to solve the 2D (vertical and radial) stratification of the disk, which is beyond our vertical one-zone approach. However, this is not expected to be crucial for the scenario depicted here. We therefore neglect the feedback of the inner JED over the outer SAD structure. In this approximation and since the resolution is computed outside-in, the temperature of the outer SAD does not depend on any assumption made on the inner JED.\\\\\n\n\nOn the contrary, the effects of the outer SAD on the inner JED are twofold and cannot be neglected. The first effect is the cooling due to advection of the outer cold material into the JED. Indeed the local advection term $q_{adv}$ can be either a cooling or a heating term, depending on the sign of the radial derivatives. This is self-consistently taken into account in our code (see Eq. (12) and Appendix A.2 in paper II).\nThe second effect results from the Compton scattering of the SAD photons on the JED electrons. This effect occurs whenever the JED is in the optically thin, geometrically thick thermal solution. This effect was not taken into account in paper II. \\\\ \n\n\nIllumination is estimated from the SAD properties, using the following geometrical prescriptions. Outside of the transition radius $R_J$, the disk luminosity is $\\displaystyle L_{SAD} = G M \\dot{M} \/ 2 R_J \\simeq 4 \\pi R_*^2 \\sigma T_s^4$, where $T_s= T_{eff}(R_*)$ is the effective temperature at the radius $R_*$ where $T_{eff}$ reaches its maximum \\citep{Frank02}. JED solutions being optically thin (or slim at worse), we assume that the radiation field in the region below $R_J$ is well described by the average energy density\n\\begin{equation} \nU_{rad} = \\omega \\frac{L_{SAD}}{4\\pi R_J^2 c} \\label{eq:Urad}\n\\end{equation}\nwhere $0 < \\omega < 1$ is a geometrical dilution factor that describes the fraction of the SAD power that irradiates the region below $R_J$ (see section~\\ref{sec:EffectOfOmega}). This applies to the bolometric luminosity, but also to the luminosity in any given energy band.\n\nThis prescription allows to compute all properties of the illumination field. This radiation is then provided to the BELM code as an external source of seed photons, and the associated cooling and reprocessed spectrum are computed. More precisely, the JED is divided in many spheres of radius $H$ in which radiation processes are computed (see paper II for more details). Here, each sphere of radius $H$ receives the power\n\\begin{equation} \nL_{s} = U_{rad} 4\\pi H^2 c = \\omega (H\/R_J)^2L_{SAD} \\label{eq:Ls}\n\\end{equation}\n\\noindent where the value of $\\omega$ is discussed in section~\\ref{sec:EffectOfOmega}.\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{EffectOfOmega.pdf}\n \\caption{Effect of the external Comptonisation of the SAD photon field on the inner JED for $\\dot{m}_{in} = \\dot{M}_{in} \/ \\dot{M}_{Edd} = 1$ and $r_J = 15$ (green vertical line on the top figures), with different dilution factors, from left to right $\\omega = 0,~1,~10,~50 \\%$. Top panels show the disk aspect as well as its Thomson optical depth in colors. Middle panels show the electron temperature as function of radius. Bottom panel displays the local spectra emitted by each radius (dashed lines) and the corresponding total disk spectrum (black solid line). For comparison, in each panel we have overplotted in gray solid lines the total spectra obtained in the other three panels. The spectra are given in Eddington fluxes $F_{Edd} = L_{Edd} \/ 4 \\pi D^2$ units for GX 339-4 (see section~\\ref{sec:GeneralPpts}). The blue lines and dots correspond to the SAD zone, while the red lines and triangles represent the JED. The white part of the spectra shows the $3-200$~keV energy range. Approximate values of the photon index $\\Gamma$ and energy cutoff $E_{cut}$, derived by comparison with a simple cutoff power-law model in this energy range, are indicated on each plot.}\n \\label{fig:EffectOfOmega}\n\\end{figure*}\n\n\\subsection{Effect of an external illumination on the JED thermal structure} \\label{sec:EffectOfOmega}\n\nAdding an outer standard accretion disk may have a colossal impact on the inner hot JED, depending on the transition radius $r_J = R_J \/ R_g$. For large values of $r_J$ (say larger than $50-100$), the power emitted by the SAD is too low to affect the global disk spectrum. But this is no longer the case when the transition radius becomes smaller. \nBesides, the geometrical dilution factor $\\omega$ used in Eq.~(\\ref{eq:Urad}) plays an important role. It is however quite tricky to obtain an accurate estimate of its value within our framework. It depends on the solid angle under which the SAD photosphere is seen by the JED and corresponds thereby to the fraction of the SAD photons that are intercepted by the JED. \n\nConsidering a spherical hot corona of radius $r_J$, centered on the black hole and embedded in an infinitely thin disk, former studies \\citep[e.g.,][]{Zdz99, Ibragimov05} led to $\\omega \\sim 2 - 25 \\%$ depending on the dynamical (\"no-torque\", \"torque\") hypothesis made at $r_J$. The inner JED accretion flow is clearly different from a sphere of radius $r_J$ (see Fig.~\\ref{fig:config}), which might naively suggest a value smaller than the above estimates. Moreover, Compton cooling also should be a function of the radius within the JED, and in the case of the geometry shown in Fig.~\\ref{fig:config}, we could expect $\\omega$ to decrease with decreasing radius. \\\\\n\n\nThere are however numerous effects that should magnify $\\omega$. \nFirst, the SAD is clearly flared, thus the infinitely thin disk approximation generally used is rather crude and tends to decrease the value of $\\omega$ \\citep[see, e.g.,][]{Meyer05, Mayer07}.\nSecond, although not considered in the litterature, the photons emitted radially by the innermost region of the SAD are also expected to radiate directly towards the JED (see Fig.~\\ref{fig:EffectOfOmega} top panel). Indeed, the photosphere $\\tau_T = 1$ necessarily crosses the disk midplane near $r_J$, allowing cold SAD photons to enter directly into the outer parts of the JED (and not only from the SAD surfaces). This should be responsible for another radiative contribution. The estimation of the corresponding photon flux emitted by the SAD and entering the JED in such a way is quite complex. It would require to solve the full (radial + vertical) radiative transfer problem to determine the photosphere properties of the SAD close to $r_J$. While this is far beyond the goal of the present paper, this effect should be similar to estimations\\footnote{Again, in the case of an infinitely thin disk, and for a disk penetrating the hot corona in $r_J$ \\citep[see again][]{Zdz99}, resulting in $\\omega$ of the order of tens of percent depending on how far the disk penetrates inside the corona.}. \nThird, the reprocessing of the X-ray emission from the JED inside the disk will also naturally increase the SAD emission \\citep{Poutanen17}. In our model, we only take into account the intrinsic disk emission, but X-ray reprocessing can be mimicked by increasing $\\omega$.\nFourth, another important effect that should be taken into account is the gravitational light bending close to the black hole. This should strongly magnify the flux of disk photons impinging the JED in comparison to the Newtonian situation where they would mainly escape away from the disk. This effect should depend on the transition radius $r_J$ as well as the radial position inside the JED. Ray tracing simulations are required here for a rigorous computation, and here again this is out of the scope of the present paper. But this effect could be the dominant one especially for small radii inside the JED or for small $r_J$, since the closer to the black hole the stronger the light bending effect. This could result in a factor of a few to be applied to the number of SAD photons entering the JED in comparison to the absence of light bending \\citep[see, e.g.,][]{Miniutti03}. All included, we believe that $\\omega$ of the order of a few tens of percent seems rather reasonable. \\\\\n\n\n\nWe report in Fig.~\\ref{fig:EffectOfOmega} the JED radial temperature distribution, as well as the corresponding spectral energy distribution (SED), for a constant $\\omega$ varying from $0$ to $50 \\%$, the two physical extremes values for this parameter. Increasing $\\omega$ obviously decreases the JED temperature and softens the SED. The variation is quite important between $\\omega=0\\%$ and $50 \\%$. Using a simple power-law model we find a spectral softening of $\\Delta \\Gamma \\simeq 0.4$ of the resulting power-law, along with a modification of its energy cutoff, from $E_{cut} \\simeq 500$~keV to $E_{cut} \\simeq 200$~keV.\n\nClearly, this effect can not be neglected, as the value of $\\omega$ has an important impact on the spectra. In this article, unless otherwise specified, we use $\\omega = 0.2$. This value appears to be close to the upper limit for previous estimations, but, considering the number of assumptions diminishing this value, we thought this was a good compromise. We note however that this does not mean that the inner JED captures $20\\%$ of the SAD luminosity, as there is still a factor $\\left( H\/R_J \\right)^2$ in Eq.~(\\ref{eq:Ls}).\n\n\n\n\n\n\\section{Synthetic observations: X-ray disk spectra and radio jet emission} \\label{sec3}\n\\label{sec:Syntheticspectra}\n\n\\subsection{From theoretical SEDs to simulated data}\n\\label{sec:Py2Xspec}\n\nThis work aims at providing synthetic hardness-intensity diagrams, or more precisely, disk fraction luminosity diagrams \\citep[DFLD, see, e.g.,][]{Kording06, Dunn10}. To that purpose, our synthetic spectra must be processed in a way similar to observational data to derive the disk and power-law components from the fits, and place the corresponding points in a DFLD. This procedure, too rarely performed, is mandatory as we intend to compare our synthetic data to observations.\n\n\nFrom our theoretical SED, an \\textsc{XSPEC} table model was first built using the \\textsc{flx2tab}\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/ftools\/caldb\/help\/flx2tab.html} command of \\textsc{ftools}\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/ftools\/}. Disk inclination was ignored for simplification but we add background and galactic absorption (\\textsc{wabs} model in \\textsc{XSPEC} with $N_H = 4 \\times 10^{21}$~cm$^{-2}$, see \\citealt{Dickey90, Dunn10, Clavel16}). Then we simulated RXTE\/PCA and HEXTE spectrum with the \\textsc{XSPEC} simulation command \\textsc{fakeit}. We use exposure times $t_{exp}$ between $1$ and $10$~ks depending on the model flux in the $3-200$~keV band in order to have a reasonably good signal-to-noise ratio. In this article, we use $t_{exp} = 10$~ks for the quiescent state (section~\\ref{sec:XrB5states}) and $t_{exp} = 1$~ks elsewhere. In Fig. \\ref{fig:Xspec}, we plot for example the simulated spectrum from the theoretical SED produced by the JED-SAD configuration with $\\omega = 0.1$ (Fig.~\\ref{fig:EffectOfOmega}, 3rd panel). \\\\\n\nEach simulated spectrum is then fitted with three different models (a): \\textsc{wabs} $\\times$ (\\textsc{cutoffpl} + \\textsc{ezdiskbb}), (b): \\textsc{wabs} $\\times$ \\textsc{cutoffpl} or (c): \\textsc{wabs} $\\times$ \\textsc{ezdiskbb}. We keep the best one according to a \\textsc{ftest} procedure \\cite[see, e.g.,][section 3.2]{Clavel16}. In the example shown in Fig.~\\ref{fig:Xspec}, model (a) gives the best fit with a reduced $\\chi^2_{red} = 248.4 \/ 213 = 1.17$. The best fit parameters are a disk blackbody temperature $T_{in} = 1.0 \\pm 0.3$~keV, a photon index $\\Gamma = 1.59 ^{+0.03}_{-0.06}$ and a lower limit in cutoff $E_{cut} > 400$~keV. These values are consistent with simple estimations performed on the theoretical data $\\Gamma = 1.64$ and $E_{cut} \\simeq 500$~keV. The corresponding best fit disk and power-law components are plotted in dashed lines in Fig.~\\ref{fig:Xspec}.\n\nThe total unabsorbed disk luminosity $L_{Disk}$ and unabsorbed power law luminosity $L_{PL}$ are computed in the $3-200$~keV range. In the example shown in Fig.~\\ref{fig:Xspec}, the powerlaw fraction, defined by $PL_f = L_{PL} \/ (L_{PL} + L_{Disk})$ is equal to $0.99 \\pm 0.01$ and the total flux in the $3-200$~keV band is $L_{tot} = L_{PL} + L_{Disk} = 2.9 \\pm 0.1 ~ \\% L_{Edd}$.\n\n\n\\begin{figure}[h!]\n \\includegraphics[width=1\\linewidth]{Py2Xspec.png}\n \\caption{Example of simulated \\textsc{RXTE\/PCA} ($3-25$~keV, in black) and \\textsc{RXTE\/Hexte} ($20-200$~keV, in red) data sets from the theoretical SED produced by the JED-SAD configuration shown in Fig.~\\ref{fig:EffectOfOmega}, third panel. The dotted lines are the power law and disk components corresponding to the best fit model. See section~\\ref{sec:Py2Xspec} for more details.}\n \\label{fig:Xspec}\n\\end{figure}\n\n\n\n\\subsection{Hard tail}\n\\label{sec:HardTail}\n\n\\begin{figure}[h!]\n \\includegraphics[width=1\\linewidth]{HardTail_Python.pdf}\n \\includegraphics[width=1\\linewidth, trim = 0.5cm 1.3cm 2.5cm 2.5cm, clip]{HardTail_Xspec.png\n \\caption{Electron temperature (top-left) and theoretical spectrum (top-right) of the configuration $\\dot{m}_{in} = 1$, $\\mu_{SAD} \\ll 1$, $\\alpha_{\\nu} = 0.1$, $r_J = r_{in}$. Each annulus is displayed as a blue dot, its associated spectrum in blue dashed lines and the total disk spectrum in black solid line. In the bottom panel, final faked and fitted data after the addition of the hard power-law tail, black for \\textsc{PCA} and red for \\textsc{\\textit{HEXTE}}. Dashed lines show the best fit obtained with \\textsc{XSPEC}, see section~\\ref{sec:HardTail} for details.}\n \\label{fig:HardTail}\n\\end{figure}\n\nIt is well know that soft states show non thermal tails generally observed above few keV \\citep{McConnell02,Remillard06}. Although uncertain, these tails are thought to be produced by a population of non-thermal electrons \\citep[see][for few investigations]{Galeev79, Gierlinski99} that are not taken into account in our model. In order to reproduce such soft states, we added a power-law component to the synthetic spectra each time the fitting procedure favors a pure blackbody emission (model (c)) and the new data were re-fitted with a modified model (c) \\textsc{wabs} $\\times$ (\\textsc{pl} + \\textsc{ezdiskbb}) model. The photon index of this power-law component is set to $\\Gamma=2.5$ and it is normalized in order to contribute to a fixed fraction of the $3-20$~keV energy range \\citep[typically between $1\\%$ and $10\\%$, ][]{Remillard06}. \\\\\n\nAn example is shown in Fig.~\\ref{fig:HardTail}. The theoretical model plotted on top of this figure corresponds to a SAD extending down to $r_J = r_{in}$ (no JED), with $\\dot{m}_{in} = 1$. A fit with an absorbed disk component (model (c)) gives a disk temperature $T_{in} = 0.97 \\pm 0.01$~keV and a total flux $L_{tot} = 4.1 \\pm 0.1 \\% L_{Edd}$ in the $3-200$~keV band. The data simulated in XSPEC include the additional power-law tail (Fig.~\\ref{fig:HardTail}, bottom panel). The best fit with new model (c) then gives $T_{in} = 0.97 \\pm 0.01$~keV, $\\Gamma = 2.5 \\pm 0.2$ and $L_{tot} = 4.5 \\pm 0.1 \\% L_{Edd}$ with a reduced $\\chi^2_{red} = 214.5 \/ 213 = 1.01$. We note that this procedure slightly increases the total flux detected in X-rays due to the addition of the hard power-law tail.\n\n\n\n\n\n\\begin{figure*}[h!\n \\centering\n \\includegraphics[width=1.\\textwidth]{Pavage_DFLD.pdf} \n \\caption{Total, disk + power-law, luminosity $L_{tot}=L_{disk} + L_{PL}$ in the $3-200$~keV energy range (in Eddington luminosity unit) is shown in function of the power-law fraction $L_{PL}\/L_{tot}$. Each point within this plot corresponds to a fully computed and then XSPEC processed hybrid JED-SAD configuration. Contours (black solid lines) are for a constant disk accretion rate $\\dot{m}_{in}$ while the color background display the disk transition radius $r_J$. Dashed black line shows the 2010-2011 cycle of GX 339-4. XSPEC fits were done with a hard tail level of $1\\%$ (left) and $10\\%$ (right). See section \\ref{sec:EffectOfRjMdot} for a description of the figure.}\n \\label{fig:EffectOfRj}\n\\end{figure*\n\n\n\n\\subsection{Jet power and radio luminosity}\n\\label{sec:Pjet}\n\nIn a JED, the jets power available is a given fraction of the accretion power\n\\begin{eqnarray}\n P_{jets} &=& b P_{acc, JED} = b \\left [ \\frac{G M \\dot{M}}{2 R} \\right ]^{R_{in}}_{R_{out}} \\nonumber \\\\\n \t\t &=& \\frac{b}{2} \\frac{\\dot{m}_{in}}{r_{in}} \\left( 1 - \\left( \\frac{r_J}{r_{in}} \\right)^{\\xi-1} \\right) L_{Edd} \\label{eq:Pjet}\n\\end{eqnarray}\nAssuming that $b$ is roughly a constant throughout an entire evolution, the jets power depends on both $\\dot{m}_{in}$ and $r_J$. We follow the computations of \\citet{Heinz03} to deduce the expected radio luminosity emitted by one jet component. It assumes that the jet emission is explained by self-absorbed synchrotron emission of non-thermal particles along the jet. We need however to modify their equations in order to account for the finite radial extent of the JED, imposing a finite radial extent of each jet. This leads to the following expression (see Appendix~\\ref{sec:JetRadio} for more details):\n\\begin{equation}\n\\frac{\\nu_R L_{R}}{L_{Edd}} \\simeq f_{R}\\, m^{\\beta-1} \\, r_{in}^{- \\frac{6p+49}{4p+16}} \\, \\dot{m}_{in}^\\beta \\, r_J^{\\frac{p+9}{p+4}}\\, \\left (1- \\frac{r_{in}}{r_J} \\right )^{\\frac{5}{p+4}}\n\\label{eq:Fr}\n\\end{equation}\nwhere $L_R= 4\\pi D^2 F_R$ is the monochromatic power emitted at the radio frequency $\\nu_R$ from an object at a distance $D$. The parameter $f_R$ is a normalization constant, $\\beta= \\frac{2p+13}{2p+8}$ and $p$ is the usual exponent of the non-thermal particle energy distribution. In the standard case with $p=2$ and $r_J$ constant, one gets the \\citet{Heinz03} dependencies, namely a radio power $\\nu_R L_{R} \\propto \\dot{m}_{in}^{17\/12}$.\\\\\n\nOur model then provides naturally both $L_R$ and $L_X$. Indeed, for any given set of parameters $(\\dot{m}_{in}, r_J)$, we can compute $L_X$ from our simulated SED, whereas an estimate of the radio luminosity can be obtained using Eq.~(\\ref{eq:Fr}). This is discussed in section~\\ref{sec4}.\n\n\n\n\n\n\\section{Reproducing typical XrB behavior: DFLD and canonical spectral states} \\label{sec4}\n\nIn this section, we show that hybrid JED-SAD configurations can, in principle, reproduce the outbursting cycles of XrBs by varying only two parameters, the disk accretion rate $\\dot{m}_{in}$ and the transition radius $r_J$. This is done in two steps. First, we need to find which ranges in $\\dot{m}_{in}$ and $r_J$ allow to cover the full DFLD. However, this diagram includes only information on X-ray emission while a cycle also deals with jet production and quenching. We thus require that the same framework reproduces radio emission at the correct level. Then, as a second step, we define five canonical spectral states characteristic of an XrB spectral evolution during an outburst and show more precisely how well our framework is able to reproduce them.\n\n\\subsection{Disk fraction luminosity diagram}\n\\label{sec:EffectOfRjMdot}\n\n\nWe perform a large parameter survey for $\\dot{m}_{in} \\in [0.01, ~ 10]$ and $r_J \\in [r_{in},~ \\sim 50 r_{in}] = [2, ~ 100]$. We compute the whole thermal structure and corresponding theoretical SED for each pair $(\\dot{m}_{in}, r_J)$, and we fit as described in section~\\ref{sec:Syntheticspectra} to get the corresponding position in the DFLD. The fits\\footnote{Only fits with $\\chi^2_{red} < 3$ have been displayed here. Few fits $(\\dot{m}_{in} \\gtrsim 5,~ r_J = r_{in})$ require the addition of a second blackbody component to better describe their spectral shape, but their position (top-left) beyond the extension of usual DFLDs makes them meaningless in the current study.} are shown in Fig.~\\ref{fig:EffectOfRj}. The smoothness of our DFLD is indicative of the absence of spectral degeneracy in our modeling (Fig.~\\ref{fig:EffectOfRj}).\nIn this figure, the mean transition radius is color-coded and the accretion rate is shown in contours of constant values. For comparison, the 2010-2011 outburst of GX 339-4 is overplotted in black dashed line\\footnote{The spectral analysis of the 2010-2011 outburst was done in the $3-25$~keV energy range (\\textsc{RXTE\/PCA}), but the models were integrated in the $3-200$~keV range and not the usual ranges \\citep[e.g.,][]{Dunn10}\\label{fnt:ranges}}.\n\n\nFig.~\\ref{fig:EffectOfRj} shows that we can cover the whole domain usually followed by XrBs within our framework. Concerning the hard states, we are able to reproduce their evolution up to high luminosities. Concerning the soft states, their position in the DFLD depends on the amplitude of the additional hard tail. With a hard tail representing $10\\%$ of the flux in the $3-20$~keV energy range throughout the cycle, we can only reproduce soft states with $L_{PL}\/L_{tot} > 0.1$ (Fig.~\\ref{fig:EffectOfRj}, right). Softer states, populating the very left part of the DFLD, require a hard tail flux fraction lower than $1\\%$ (Fig.~\\ref{fig:EffectOfRj}, left).\n\nAs expected, the accretion rate is mainly responsible for the global X-ray luminosity of the system, leading to almost horizontal isocontours for $\\dot{m}_{in}$ in the DFLD. Indeed, the higher the accretion rate, the higher the total available accretion energy $P_{acc} \\propto \\dot{m}_{in}$. However, part of the energy can be advected, which explains the sharp variations of the isocontours at high luminosities (see discussion below). The effect of the transition radius $r_J$ follows the predictions of paper I. At large transition radii, most of the emission originates from the JED, as the outer SAD has no detectable influence. These solutions display power-law spectra for all accretion rates (see paper II). This is the reason why they appear on the right-side of the DFLD. At small transition radii, two effects appear in the RXTE energy range of our simulated data. First, as its temperature and flux increase with decreasing $r_J$, the SAD blackbody emission starts appearing in the SED around $3$~keV. Second, the closer the SAD, the stronger its illumination becomes, cooling down the inner JED and producing softer spectra. Combining these two effects leads to a disk dominated spectrum, with a power-law fraction becoming entirely dominated by the high energy tail when $r_J \\rightarrow r_{in}$. \\\\\n\n\n\\subsection{$L_X$ dependencies on $\\dot{m}_{in}$}\n\n\n\\begin{figure}[h\n \\centering\n \\includegraphics[width=\\columnwidth]{LbolLx39_vs_mdot.pdf}\n \\caption{Bolometric (top) and $3-9$~keV (bottom) luminosities in function of the mass accretion rate $\\dot{m}_{in}$ onto the black hole. This plot is extracted from Fig.~\\ref{fig:EffectOfRj}, done with a $10\\%$ hard tail (right). The colors are for different values of the transition radius $r_J$. Four different $L \\propto \\dot{m}_{in}^{\\alpha}$ regimes are shown. Also, the $r_J = r_{in}$ has been drawn in dashed black to be visible at low accretion rate in the bottom panel.}\n \\label{fig:Lxmdot}\n\\end{figure\n\n\n\nFigure~\\ref{fig:Lxmdot} shows the bolometric ($L_{bol}$, top) and $3-9$~keV ($L_{3-9}$, bottom) luminosity deduced from our synthetic SED as a function of the accretion rate at the inner radius $\\dot{m}_{in}$. The colors correspond to different transition radius $r_J$.\nThis figure illustrates different concerns about the radiative efficiency of accretion flows. \\\\\n\n\n\n\nOn the top panel and in the SAD mode ($r_J = r_{in} = 2$), the bolometric luminosity follows the radiatively efficient regime $L_{bol} \\propto \\dot{m}_{in}$ as long as $\\dot{m}_{in} < 5$. Below this accretion rates, the SAD is indeed radiating all of its available energy. At $\\dot{m}_{in} > 5$, the SAD enters the slim domain, where more and more energy becomes advected instead of being radiated away. As a consequence, the global luminosity has a steeper slope $L_{bol} \\propto \\dot{m}_{in}^{0.5}$.\n\n\nAs $r_J$ increases, the JED mode starts to have a bigger extent, and at any given accretion rate $\\dot{m}_{in}$ the luminosity decreases as $r_J$ increases. This is the result of two different effects. First, as $r_J$ increases, more and more energy is controlled by the JED and transferred to the jets $b = P_{jets} \/ P_{acc, JED} = 0.3$ (papers I and II) instead of being radiated. Second, the JED thermal equilibrium is often strongly affected by advection, as $f = P_{adv} \/ P_{acc, JED} \\propto \\varepsilon^2$ is no more negligible (paper II). Combining these two effects, a more representative formulation is $L_{bol} \\propto (1-b-f) ~ \\dot{m}_{in}$, where $f \\propto \\varepsilon^2 = \\varepsilon^2 (\\dot{m}_{in})$ also is a function of accretion rate.\n\nAt low accretion rates $\\dot{m}_{in} < 0.5$, the JED is optically thin and geometrically thick with $\\varepsilon \\simeq 0.2-0.3$ (termed thick disk solution in paper II). In the thick disk branch, the low density of the plasma allows $q_{ie} \\propto n_e^{2}$ to be negligible. Ions are neither cooled down by radiation nor by Coulomb interactions: $T_i \\gg T_e$. Contrarily to usual one-temperature plasmas, the disk thickness is then only linked to the ion pressure, $P_{gas,i} \\gg P_{gas,e} + P_{rad}$, leading to $q_{adv} \\simeq q_{adv,i} \\gg q_{adv,e}$ (see Eq. (13) in paper II). In the ion thermal equilibrium from Eq. (\\ref{eq:IONS}), advection is directly determined by ion heating, $q_{adv} \\simeq q_{adv,i} \\simeq (1-\\delta) q_{turb}$, leading to $f = q_{adv} \/ q_{turb} \\simeq 0.5$. Since $q_{adv,e} \\ll q_{adv,i} \\simeq (1-\\delta) q_{turb}$, we obtain $q_{adv,e} \\ll \\delta q_{turb}$. In the electron equilibrium Eq. (\\ref{eq:ELECTRONS}), radiation is then determined by $q_{rad} \\sim \\delta q_{turb} \\propto \\dot{m}_{in}$, leading to the trend $L_{bol} \\propto \\dot{m}_{in}$, unexpected in a thick disk\\footnote{This in only true if $(1-\\delta) \\sim \\delta$, which is the case here. In ADAFs, where $\\delta = 1\/2000$, even if $q_{adv,e} \\ll (1-\\delta) q_{turb}$ the factor $\\delta \\ll 1$ would not ensure that $q_{adv,e} \\ll \\delta q_{turb}$, and this reasoning would not stand.}. In the end, combining the loss of power through jets ($b$) and in advection ($f$) reduces the JED luminosity. For instance, for $r_J = 50$ (yellow color) it is reduced by a factor $1 \/ (1-b-f) \\sim 5$ compared to the SAD mode power.\n\nAt high accretion rates $\\dot{m}_{in} > 2$, the JED mode is optically thick and geometrically slim with $\\varepsilon \\simeq 0.1$ (termed slim disk solution in paper II). The bolometric luminosity now has a slope $L_{bol} \\propto \\dot{m}_{in}^{0.5}$ (see the previous discussion on the slim SAD mode), with a factor $\\sim 2$ times lower in luminosity. Indeed, in this case, $f \\simeq 0.15$, due to the lower temperature (and then the lower $\\varepsilon$) compared to the thick disk solution.\n\nBetween these two solutions, from the thick to the slim disks, $f$ decreases from $0.45$ to $0.15$. The disk radiates more and more energy and the luminosity increases until it reaches the $L_{bol} \\propto \\dot{m}_{in}^{0.5}$ slope. During this transition, the slope is more abrupt and fits well with $L_{bol} \\propto \\dot{m}_{in}^{1.5}$. \\\\\n\n\nThe bottom panel of Fig.~\\ref{fig:Lxmdot} shows that the $L_{3-9}$ variation with the accretion rate is different from $L_{bol}$. In the SAD mode ($r_J = r_{in} = 2$), $L_{3-9} \\propto \\dot{m}_{in}^{\\sim 2}$ for $\\dot{m}_{in} < 1$ and it slowly drops down to $L_{3-9} \\propto \\dot{m}_{in}^{0.5}$ while approaching the slim region. The JED mode with $r_J = 50$ remains closer to the bolometric behavior, with still $L_{3-9} \\propto \\dot{m}_{in}$ when $\\dot{m}_{in} < 0.5$ and roughly $L_{3-9} \\propto \\dot{m}_{in}^{1.5}$ during the transition and in the slim region. At very low accretion rates, the JED mode radiates more energy in the $3-9$~keV band up to a factor $\\sim 4$, while at higher accretion rates the SAD can radiate as high as $\\sim 17$ times more energy in this range. \\\\\n\n\nTwo interesting comments can be done from Fig.~\\ref{fig:Lxmdot}. First a \"radiatively efficient accretion flow\" does not necessarily mean $L \\propto \\dot{m}_{in}$. Indeed, as shown before in the JED dominated mode ($r_J = 50$) we find $L \\propto \\dot{m}_{in}$, while the JED radiates only few tens of its total available energy. The term \"radiatively efficient\" seems therefore inappropriate. Second, the evolution of the luminosity with the accretion rate strongly depends on the energy range used. In the SAD mode, while the disk is indeed radiatively efficient and $L_{bol} \\propto \\dot{m}_{in}$, the $3-9$~keV luminosity follows a $L_{3-9} \\propto \\dot{m}_{in}^{1.5-2}$ regime, which could be considered as the signature of a radiatively inefficient flow. This clearly means that the interpreation of the luminosity variation with the accretion rate in terms of radiative efficiency can be strongly misleading and should be done with caution.\\\\\n\n\n\n\n\n\nFinally, Fig.~\\ref{fig:Lxmdot} also illustrates that the functional dependence $L_{3-9} (\\dot{m}_{in})$ can be much more complex than a single power law. This is quite promising as it is known that $L_{3-9} (\\dot{m}_{in})$ may need to vary from one object to another \\citep[][section~4.3.3 and Fig.~7]{Coriat11}. However, we cannot go further without considering a proper outbursting cycle. For instance, Fig.~\\ref{fig:EffectOfRj} clearly shows that, in order to successfully reproduce the 2010-2011 cycle of GX 339-4 (black lines in Fig~\\ref{fig:DFLD_complet}), one would need to (1) rise up, namely increase $\\dot{m}_{in}$ from the quiescent state until the highest hard state; then (2) transit left by decreasing the transition radius $r_J$ until the full disappearance of the JED; (3) drop down in the soft realm by decreasing $\\dot{m}_{in}$ and finally (4) transit right back to the hard zone by increasing $r_J$. It can thus be inferred from Fig.~\\ref{fig:Lxmdot} that the evolution in time of the X-ray luminosity is sharper than $L_{3-9} \\propto \\dot{m}_{in}$ at high luminosities, because of the necessary decrease of $r_J$. A detailed modelling of the actual track followed by GX 339-4 during a full cycle will be presented in a forthcoming paper.\n\n\n\n\\subsection{Radio fluxes} \\label{sec:RadioFluxes}\n\n\n\\begin{figure}[h!\n \\centering\n \\includegraphics[width=\\columnwidth]{LradioLx.pdf}\n \\caption{Power in the 8.6 GHz radio band in function of the 3-9 keV X-ray power (both in Eddington luminosity). Black solid lines are for constant transition radius $r_J$, while the color background shows the accretion rate $\\dot{m}_{in}$. This figure has been done using a similar procedure than in Fig.~\\ref{fig:EffectOfRj}. The black points are the observed values for GX 339-4 during its cycles between 2003 and 2011. A possible theoretical jet line has been drawn in red (see section~\\ref{sec:RadioFluxes}).}\n \\label{fig:PJET}\n\\end{figure\n\n\n\n\n\nFor any given hybrid JED-SAD disk configuration, computed with a pair of parameters ($\\dot{m}_{in}, r_J$), one can also derive an estimation of the one-sided jet radiative power $P_R$ emitted at the radio frequency $\\nu_R = 8.6$~GHz. Assuming an electron distribution with $p=2$, Eq.~(\\ref{eq:Fr}) leads to\n\\begin{equation}\nP_R \\equiv \\nu_R L_R = \\tilde f_R \\, \\dot{m}_{in}^{17\/12} r_J \\left ( r_J - r_{in} \\right )^{5\/6} \\, L_{Edd}\n\\label{eq:PR}\n\\end{equation}\nwhere the dimensionless factor $\\tilde f_R$ incorporates factors such as $m$ or $r_{in}$, see Appendix \\ref{sec:JetRadio}. For the sake of simplicity, we assume that this factor is a constant throughout the whole cycle. This is far from being obvious, but our goal here is simply to show that hybrid configurations have the potential to reproduce simultaneously both X-rays and jet radio emission. \nTo have an estimate of $\\tilde f_R$, we require that two radio observations at $\\nu_R= 8.6$~GHz, one during the quiescent state and the other during the high-luminosity hard state (see their definition below), are qualitatively reproduced. We obtain $\\tilde f_R = 3 \\times 10^{-11}$. This allows us to compute the radio power $P_R$ as a function of ($\\dot{m}_{in}, r_J$) and finally relate the radio power to the observed $3-9$~keV power.\nIn Fig.~\\ref{fig:PJET}, the same procedure than in Fig.~\\ref{fig:EffectOfRj} is used, but with binning of the true integrated luminosity $L_{3-9~\\text{keV}}$ and radio power $P_R$. In addition, radio and X-ray observations of GX 339-4 were overplotted in black dots. It can be seen that most of them correspond to $r_J \\sim 10-50$ while the accretion rate spans $\\dot{m}_{in} \\sim 0.01$ to almost $5$. Thus, reproducing the observed radio\/X-ray diagram requires variations in mass accretion rate, from $\\dot{m}_{in} < 0.1$ to almost $5$ here. However, in order to describe the disappearance or reappearance of the steady radio emission, i.e. the crossing of the jet line \\citep{Fender04}, one needs to invoke variations in transition radius: a decrease in $r_J$ when the jet is quenched (bottom-right), and an increase when the jet re-appears (top-left). This is very promising for the model to reproduce observations, but further investigation needs to be done.\n\n\n\\begin{table*}[h!]\n\\centering\n{\\renewcommand{\\arraystretch}{1.3}\n \\begin{tabular}{c | c c c c | c c | c c c c c}\n & \\multicolumn{4}{c|}{Typical observed states} & \\multicolumn{2}{c|}{Parameters} & \\multicolumn{5}{c}{Results of fits} \\\\ \\hline\nSpectral & $\\mathbf{PLf}$ & $\\textbf{X-rays}$ & $\\mathbf{\\Gamma}$ & $\\textbf{Radio}$ & $\\mathbf{\\dot{m}_{in}}$ & $\\mathbf{r_J} $ & $\\mathbf{PLf}$ & $\\textbf{X-rays}$ & $\\mathbf{\\Gamma}$ & $\\textbf{Radio}$ & $ \\chi^2_{red}$\\\\\nstate & & $(\\% L_{Edd})$ & & (mJy) & $(L_{Edd}\/c^2)$ & $(R_g)$ & & $(\\% L_{Edd})$ & & (mJy) & \\\\ \\hline\n\\textbf{Q} & $ 1 $ & $ < 0.1 $ & $ 1.5-2.1 $ & $ 1 $& $ 0.06 $ & $ 100 $ & $ 1 $ & $ 0.11 \\pm 0.01 $ & $ 1.9^{+0.3}_{-0.3}$ & $ 2.8 $ & $0.96$ \\\\\n\\textbf{LH} & $ 1 $ & $ 1 $ & $ 1.5-1.6 $ & $ 5 $ & $ 0.4 $ & $ 50 $ & $ 1 $ & $ 0.92 \\pm 0.05 $ & $ 1.64^{+0.04}_{-0.04} $ & $ 11.5 $ & $1.10$ \\\\\n\\textbf{HH} & $ 1 $ & $ >10 $ & $ 1.6-1.8 $ & $ 25 $& $ 2 $ & $ 15 $ & $ 0.98 \\pm 0.02 $ & $ 13.1 \\pm 0.1 $ & $ 1.68^{+0.03}_{-0.02} $ & $ 20.2 $ & $1.60$ \\\\ \n\\textbf{HS} & $ < 0.3 $ & $ 5 $ & $ 2-3 $ & $ < 0.01 $& $ 0.75 $ & $ 2 $ & $ 0.13 \\pm 0.05$ & $ 4.6 \\pm 0.1 $ & $ 2.4^{+0.3}_{-0.7} $ & $ 0 $ & $0.90$ \\\\\n\\textbf{LS} & $ < 0.3 $ & $ 1 $ & $ 2-3 $ & $ < 0.01 $& $ 0.45 $ & $ 2 $ & $ 0.2 \\pm 0.05 $ & $ 1.40 \\pm 0.04 $ & $ 2.5^{+0.3}_{-0.6} $ & $ 0 $ & $1.04$ \\\\ \\hline\n\\end{tabular}}\n \\caption{Typical observed properties of the five canonical states (left), pairs of parameters ($\\dot{m}_{in}$ and $r_J$, center) and XSPEC fits (right) results associated to the five chosen canonical states Q, LH, HH, HS and LS. Fits are performed using the simple model described in section~\\ref{sec:Py2Xspec}, and the definition of the five states is detailed in section~\\ref{sec:XrB5states}.}\n \\label{table:States}\n\\end{table*}\n\n\n\\subsection{XrB canonical spectral states} \\label{sec:XrB5states}\n\n\nIn section \\ref{sec:EffectOfRjMdot}, it is shown that hybrid JED-SAD configurations can cover the observed DFLDs by varying independently $\\dot{m}_{in}$ and $ r_J$. \nWe focus here on the five typical spectral states that any given XrB needs to cross (or get close to) when making a full cycle, and detail their characteristics in our JED-SAD framework. These five states are shown in Fig.~\\ref{fig:DFLD_complet} and are named quiescent state (hereafter Q), low-luminosity hard state (LH) and low-luminosity soft state (LS), both at the soft-to-hard lower transition branch, and high-luminosity hard state (HH) and high-luminosity soft state (HS) both at the hard-to-soft upper transition branch.\n\n\\begin{figure}[h!]\n\\centering\n\\sidecaption\n \\includegraphics[width=1.\\columnwidth]{b1_4.pdf}\n \\caption{DFLDs for the past outbursts of GX 339-4 between MJD$50290$ and MJD$55650$ extrapolated in the $3-200$~keV energy range\\footref{fnt:ranges}. A typical cycle goes from Q (black), and crosses LH (orange) up to HH (red). It then transits to the HS (blue), decreases to LS (cyan) until it transits back to LH, before decreasing down to Q. The stars mark the positions of the five canonical spectral states Q, LH, HH, HS and LS defined in section~\\ref{sec:XrB5states}.}\n \\label{fig:DFLD_complet}\n\\end{figure}\n\n\n\n\\begin{figure*}[h!\n \\centering\n \\includegraphics[width=1.\\textwidth]{TypicalStates.pdf}\n \\caption{Computed radial structures of hybrid JED-SAD disk configurations associated to the canonical states defined in Table~\\ref{table:States}. From left to right: \\textbf{Q} for quiescent state, \\textbf{LH} for low-hard, \\textbf{HH} for high-hard, \\textbf{HS} for high-soft and \\textbf{LS} for low-soft. Top: electron temperature $T_e$ at the disk mid plane. Bottom: Thomson optical depth $\\tau_T$. Red triangles show the JED zone, and blue dots describe the SAD zone. The vertical green line marks their separation at the transition radius $r_J$. The vertical yellow line marks the transition from a gas to radiation pressure supported regime within the SAD. In addition, the other two possible thermal solutions are shown in grey when present, the unstable in circles and the thin disk in triangles, see paper II.}\n \\label{fig:TypicalStates}\n\\end{figure*\n\n\nThe quiescent state chosen is clearly not the most quiescent state that can be reached by an XrB. It is at position in the DFLD that any object needs to cross while going up and down. Our choice of the two soft states in the DFLD is somewhat arbitrary as it depends on the chosen level of the hard tail, see section~\\ref{sec:HardTail}. Our computed spectra are also shown from $0.5$ to $500$~keV but remember that the observational PCA and HEXTE data are available only from $3$ to $200$~keV. \\\\\n\n\n\n\nTable~\\ref{table:States} middle panel shows the values of the parameters $\\dot{m}_{in}$ and $r_J$ that better characterize these five canonical states. We also have reported in this table the values (and their associated $3 \\sigma$ errors) of the power-law fraction $PLf$, X-ray luminosity and spectral index $\\Gamma$ derived from XSPEC fits, as well as the expected radio flux density at $8.6$~GHz, $F_{8.6~\\text{GHz}}$. These fluxes have been computed in mJy using Eq.~(\\ref{eq:PR}), namely $F_{8.6~\\text{GHz}}= 10^{26} \\times P_R\/(4\\pi D^2 \\nu_R)$, with $\\nu_R= 8.6$~GHz and $\\tilde{f}_R = 3 \\times 10^{-11}$. Figs.~\\ref{fig:TypicalStates} and \\ref{fig:SpectralStates} illustrate the thermal state radial distribution (temperature, optical depth) and the theoretical and faked spectra. An accurate representation of the physical structure (size) and temperature (color) of the disk in those five states is shown in Fig.~\\ref{fig:JoliesImages}.\n \n\\begin{figure*}[h!\n \\centering\n \\includegraphics[width=1.\\textwidth]{SpectralStates.pdf}\n \\caption{Theoretical SED (top) and XSPEC spectral fits (bottom) for the five canonical spectral states computed in Fig.~\\ref{fig:TypicalStates}. It can be seen that hard states spectra are always dominated by the comptonization of soft photons, mostly due to local Bremsstrahlung and cold photons from the outer SAD. The white area in the theoretical SED corresponds to the observationally relevant $3-200$~keV energy band. The value of the spectral index $\\Gamma$ is shown for each state, with its errors.}\n \\label{fig:SpectralStates}\n\\end{figure*\nIn the following, we discuss each of the canonical states with more details. \\\\% A summary of the canonical states can be found in Table~\\ref{table:TypicalStates}, and the results of the resolution and fits shown as examples are presented in Fig.~\\ref{fig:TypicalStates} (temperature and optical depth), Fig.~\\ref{fig:SpectralStates} (spectra) and Table~\\ref{table:States} (fits).\n\n\n\\noindent {\\bf - Q state:} the quiescent state has an X-ray luminosity lower than $0.1 \\%$ Eddington, with a typical power-law spectrum of index $\\Gamma \\simeq 1.5-2.1$ in the observed $3-200$~keV band \\citep{Remillard06}. It exhibits faint but steady radio and IR luminosities fluxes \\citep{Fender01, Corbel13}, probing weak but detectable jets. \nIt is located at the bottom-right of the DFLD. In our JED-SAD framework it is characterized by a very low accretion rate $\\dot{m}_{in} \\lesssim 1$ and a relatively high transition radius $r_J \\gg r_{in}$. In the example shown in this section we choose $\\dot{m}_{in} = 0.06$ and $r_J = 100$. The innermost region of the disk (from $r \\simeq 30$ down to $r_{in}$) is optically thin with electron temperature as high as $T_e \\simeq 10^{10}$~K (Fig.~\\ref{fig:TypicalStates}, left panel). This results in a global spectrum that is the sum of multiple power-law spectra with roughly the same shape $\\Gamma \\sim 1.6-2$ and $E_{cut} \\gg 200$~keV (Fig.~\\ref{fig:SpectralStates}, left panel), in good agreement with the observations. In addition, the power available in the jets is relatively small due to a very low accretion rate (Table \\ref{table:States}).\\\\\n\n\n\n\n\\noindent {\\bf - LH state:} the low-luminosity hard state is characterized by a power-law dominated spectrum, with spectral index $\\Gamma \\simeq 1.5-1.6$, no cutoff detected $E_{cut} > 200$~keV \\citep[][and references therein]{Grove98, Zdz04} and a typical luminosity $L_{tot} \\simeq 1 \\% L_{Edd}$. This state is also associated to steady and high radio and IR luminosities suggesting powerful jets. In this article we choose $\\dot{m}_{in} = 0.4$ and $r_J =50$. As shown in Fig.~\\ref{fig:TypicalStates} top panel, the temperature of this flow increases quickly from $T_e \\simeq 10^{6}$~K in the outer parts of the JED to $T_e \\gtrsim 5 \\times 10^{9}$~K in the inner parts for most of the JED extension (from $r = 20$ down to $r_{in}$). This state does not strongly depend on $r_J$ as long as it is larger than a few tens of $r_{in}$, as the global spectrum is the sum of similar spectra with $\\Gamma \\sim 1.2-1.8$ (Fig.~\\ref{fig:SpectralStates}, LH-panel). This configuration is also accompanied by more powerful jets, due to larger accretion rate compared to Q-states, in agreement with stronger observed radio emission. \\\\\n\n\\noindent {\\bf - HH state:} the high-luminosity hard state is also characterized by a power-law dominated spectrum, with a spectral index $\\Gamma \\simeq 1.6-1.8$, but with a high energy cutoff generally detected $E_{cut} \\simeq 50-200$~keV \\citep{Motta09} and luminosities as high as $30 \\% L_{Edd}$ \\citep{Dunn10}. Actually, as the luminosity increases, $E_{cut}$ is observed to decrease from $>200$~keV to $\\sim 50$~keV, while the spectral index slightly changes from $\\Gamma = 1.6$ to $\\Gamma = 1.8$ before transiting to the soft state \\citep[see Figure 6 in][]{Motta09}. Those states also show the highest radio and IR fluxes \\citep{Coriat09}, suggesting the most powerful jets of the cycle. In our JED-SAD framework this state is characterized by a larger accretion rate $\\dot{m}_{in} > 1$, we choose $\\dot{m}_{in} = 3$. As shown in paper II, at such high accretion rate the hot geometrically thick disk solution switches to a denser and cooler solution, the so-called slim disk. The disk is optically slim $\\tau \\sim 1-10$ and rather warm $T_e \\sim 10^{8-9}$~K (Fig.~\\ref{fig:TypicalStates}, middle panel). The spectra associated to those slim disks solutions are closer to a very hot multi-temperature disk blackbody emission (paper II). The combination between electron temperature and optical thickness distribution with radius produces a spectral shape in agreement with observations (i.e. $\\Gamma \\simeq 1.6-1.8$ and $E_{cut} \\in [50, ~ 200]$~keV). Once we choose $\\dot{m}_{in} = 3$, the range of appropriate values for transition radius is rather narrow to reproduce the value of $\\Gamma$, we adopt $r_J = 15$. The large $\\dot{m}_{in}$ and $r_J$ result in a large jet power consistent with observations (see Table~\\ref{table:States}).\n\n\\noindent {\\bf - HS state:} the high-luminosity soft state is defined by a dominant multi-temperature blackbody with maximum effective temperature $T_{in} \\lesssim 1$~keV and total flux $L_{tot} \\sim 5 \\% L_{Edd}$. In addition, many of the soft states display a minor component, the hard tail (see section \\ref{sec:HardTail}). These states are also characterized by the absence of steady radio emission, interpreted as the jet disappearance \\citep[][]{Fender99, Corbel00}. In our JED-SAD framework this translates to $r_J = r_{in}$, i.e. our accretion flow is entirely in a SAD mode. In the example shown in Fig.~\\ref{fig:TypicalStates}, we choose $\\dot{m}_{in} = 0.75$ with a $10\\%$ hard tail. \\\\\n\n\\noindent {\\bf - LS state:} the spectral shape of the low-luminosity soft state is similar to the HS, with a typical maximum effective disk temperature $T_{in} \\gtrsim 0.6-0.7$~keV, a $\\Gamma \\simeq 2-3$ hard tail, and total luminosity $\\sim 5$ times lower. \nWe define our canonical LS state with the same level of hard tail. In our JED-SAD framework, this correspond to lower $\\dot{m}_{in}$ but still $r_J = r_{in}$ (no JED). In the example shown in this section, we choose $\\dot{m}_{in} = 0.45$. The absence of JED means there are no jets, i.e. no radio emitted. \\\\\n\n\n\n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=1.\\columnwidth]{b2.pdf}\n \\caption{Computed geometrical shape of the hybrid disk, consistent with the dynamical resolution (Fig.~\\ref{fig:TypicalStates}) and SED (Fig.~\\ref{fig:SpectralStates}) for each of the five canonical states. The color background is the central electron temperature. Note that the X-scale is logarithmic and the Y-scale is linear.}\n \\label{fig:JoliesImages}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Concluding remarks} \n\\label{sec:Ccl}\n\nIn paper II, we studied the thermal equilibrium of jet-emitting disks (JED). JEDs are assumed to be thread by a large scale vertical magnetic fields, building two jets that produce a torque responsible for supersonic accretion.\n\nIn this article, we extend the code to compute thermal equilibria of hybrid disk configurations. This configuration assumes an inner JED and an outer standard accretion disk (SAD), characterized by a highly subsonic accretion speed. The transition between those two flows is assumed to be abrupt ($\\Delta R \/ R \\ll 1$) at some transition radius $r_J$. As argued in section~\\ref{sec:inter}, such a transition requires a discontinuity in the disk magnetization $\\mu$ that can be obtained if the transition radius $r_J$ is a steep density front. The transition radius $r_J$ would thus correspond to a density front advancing or receding within the disk during an outburst, as also found in the context of ADAF-SAD transitions \\citep{Honma96, Manmoto00}. Why such a density front would be present is an open question, possibly answered by how matter is initially brought in towards the disk inner regions. In any case, if such a front is indeed produced, it is not clear how it would be maintained over the long duration of the outburst.\n\nNow, such a density front is known to be favorable to the Rossby wave instability \\citep[][and references therein]{Tagger99, Lovelace99, Li00, Tagger04, Meheut10}, which leads to the formation of non-axisymmetric vortices within the disk. Whether or not the density front is smeared out and destroyed or simply perturbed (leading possibly to quasi-periodic oscillations) remains to be investigated. We refer the interested reader to the discussion on timing properties in paper I, section 4.\n\nOn the other hand, one might argue as well that such a discontinuity in the disk magnetization is unrealistic and that, instead, there is a continuous increase in $\\mu$ towards the disk inner regions \\citep{Petrucci08}. Assuming that such a situation would be indeed realized, the transition radius $r_J$ required in our spectral calculations would then be interpreted as the transition from the outer optically thick disk to the inner optically thin disk. Correspondingly, one could argue that the outer low-magnetized disk regions would give rise to winds whereas jets would be launched from the inner highly magnetized disk regions (JED). The difficulty with this scenario is that it relies on the disk mass loss and the radial distribution of the large scale vertical magnetic field, both unknown to date. Our simple approach, that assumes a sharp JED-SAD radial transition, can be seen as a first step towards addressing this difficult topic in XrB accretion disks. \\\\\n\n\n\nThe outside-in radial transition in accretion speed translates thermally, from an outer optically thick and cold accretion flow to an inner optically thin and hot flow. The soft photons emitted by the outer disk also provide a non-local cooling term which, added to advection of internal energy, allow a smooth thermal transition between these two regions. For a given JED-SAD dynamical solution the corresponding spectrum depends only on the mass accretion rate onto the black hole $\\dot{m}_{in}$ and the transition radius $r_J$ between the two flows. We explore in this article a large range in $\\dot{m}_{in}$ and $r_J$. Using XSPEC, we build synthetic spectra and fit them using a standard observers procedure (section~\\ref{sec:Py2Xspec}), allowing us to easily compare the resulting fits to observations.\n\nWe show that this framework is able to cover the whole domain explored by typical cycles in a disk fraction luminosity diagram (Fig.~\\ref{fig:EffectOfRj}). Furthermore, five canonical X-ray spectral states representative of a standard outburst are quantitatively reproduced with a reasonable set of parameters (Figs.~\\ref{fig:TypicalStates} to \\ref{fig:JoliesImages} and Table~\\ref{table:States}). A very interesting and important aspect of this framework is its ability to simultaneously explain both X-ray and radio emissions (Fig~\\ref{fig:PJET} and Table~\\ref{table:States}). In a forthcoming paper we will show the required time sequences $\\dot{m}_{in}(t)$ and $r_J(t)$ needed to reproduce a full cycle within the JED-SAD paradigm. \\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgements}\n We are grateful to the anonymous referee for his\/her careful reading of the manuscript. The authors acknowledge funding support from French Research National Agency (CHAOS project ANR-12-BS05-0009 http:\/\/www.chaos-project.fr), Centre National de l'Enseignement Superieur (CNES) and Programme National des Hautes Energies (PNHE) in France. SC is supported by the SERB National Postdoctoral Fellowship (File No. PDF\/2017\/000841).\n\\end{acknowledgements}\n\n\n\\newpage \n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\IEEEPARstart{F}{or} a given $q{\\in}\\mathbb{N}$, define a set of complex exponential sequences as $H_q = \\{S_{q,k}(n) = e^{\\frac{j2{\\pi}kn}{q}}|0{\\leq}k{\\leq}q-1, (k,q)=1\\}$, where $0{\\leq}n{\\leq}q-1$ and $(k,q)$ denotes the greatest common divisor (gcd) between $k$ and $q$. \nBy adding all the elements of $H_q$, mathematician Srinivasa Ramanujan introduced a trigonometric summation called as Ramanujan Sum (RS) in $1918$, denoted as $c_q(n)$ \\cite{Ramanujan}.\nLater in 2014, P. P. Vaidyanathan introduced a finite length signal representation using $c_q(n)$ and its circular shifts \\cite{6839014}, \\cite{6839030}.\n\n\n\nMotivated by this, we introduced a trigonometric summation known as Complex Conjugate Pair Sum (CCPS) in one of our previous works \\cite{Shah}. \nAs $(k,q)$ $=$ $(q-k,q)$, for every $S_{q,k}(n){\\in}H_q$, there exists a complex conjugate sequence $S_{q,q-k}(n){\\in}H_q$, both together form a complex conjugate pair. \nCCPS $(c_{q,k}(n))$ is defined by adding each complex conjugate pair, i.e.,\n\\begin{equation}\nc_{q,k}(n) = 2Mcos\\left(\\frac{2{\\pi}kn}{q}\\right),\n\\end{equation}\nwhere \\footnotesize$M=\\begin{cases}\n\\frac{1}{2}&\\ \\text{if } q=1\\text{ (or) }2\\\\\n1&\\ \\text{if } q{\\geq}3 \n\\end{cases}$\\normalsize, $k{\\in}{\\hat{U}_q}$ if $q>1$ and $k=1$ if $q = 1$, refer \\textit{Notations} for ${\\hat{U}_q}$. \nRecently, S. W. Deng et al., introduced a two-dimensional subspace spanned by a complex conjugate pair $\\{S_{q,k}(n),S_{q,q-k}(n)\\}$ known as Complex Conjugate Subspace (CCS) \\cite{7544641}, denoted as $v_{q,k}$. \nIn \\cite{Shah}, we provided a new basis for CCS using CCPS.\nFurther, a finite length signal is represented as a linear combination of signals which belong to CCSs known as Complex Conjugate Periodic Transform (CCPT) \\cite{Shah}.\n\nInspired from \\cite{6839014,6839030,7964706}, in this letter we discuss several properties of CCPSs and CCSs, which may find applications in signal processing.\nContributions of this letter can be divided into two parts. \nIn the first part, we show that\nthe operation of linear convolution between a given signal $x(n)$ and $\\bar{c}_{q,k}(n)$ is equivalent to computing the first derivative of $x(n)$, where $\\bar{c}_{q,k}(n)$ denotes one period data of ${c}_{q,k}(n)$. \nThis operation is also equivalent to the second derivative if we consider an odd number $q$ and a circular shift of $\\frac{q-1}{2}$ for $\\bar{c}_{q,k}(n)$. \nThen, the problem of edge detection in an image is addressed using this derivative equivalent operation. \nMoreover, we compare these results with the results obtained by using RSs.\n\nIn the second part, we prove the following properties:\n\\begin{itemize}\n\\item CCS is a shift-invariant subspace.\n\\item Since CCPT is a non-orthogonal transform \\cite{Shah}, we\ncompute the projections onto CCSs. \nFurther, we show that the projection matrix is a circulant matrix, which reduces the computational complexity of projections.\n\\item CCS is closed under circular cross-correlation. Moreover, the circular auto-correlation of any finite length sequence is equal to the weighted linear combination of its projections auto-correlation.\n\\end{itemize}\n\n\nThe structure of this letter is as follows: Using CCPSs as derivatives and the problem of detecting edges in an image are studied in Section \\RNum{2}.\nProperties of CCSs are discussed in Section \\RNum{3}. Finally, conclusions are drawn in Section \\RNum{4}.\n \n\\textit{Notations:}\n$(a,b)$ indicates the gcd between $a$ and $b$. A least common multiple is denoted as $lcm$.\nRounding the value $a$ to the greatest integer less than or equal to $a$ is denoted as $\\floor*{a}$.\nFor a given $n\\in\\mathbb{N}$, Euler's totient function $\\varphi(n)$ is defined as $\\varphi(n) = \\sum\\limits_{i=1}^{n}\\floor*{\\frac{1}{(i,n)}}$. As $(k,n) = (n-k,n)$, $\\varphi(n)$ is even for $n\\geq 3$. \nSymbol $d|N$ denotes that $d$ is a divisor of $N$. \nDefine a set $\\hat{U}_n = \\{a{\\in}\\mathbb{N}\\ |\\ 1{\\leq}a{\\leq}\\floor*{\\frac{n}{2}}, (a,n)=1\\}$, hence $\\#\\hat{U}_n = \\frac{\\varphi(n)}{2}$.\n$M_{m,n}(\\mathbb{C})$ indicates set of all $m\\times n$ matrices with entries from complex numbers. If $m=n$, $M_{m,n}(\\mathbb{C})=M_n(\\mathbb{C}).$\n\\section{CCPSs as Derivatives and Application}\nHere we perform an operation using CCPSs, which is equivalent to the derivative. In particular, we prove the following:\n\\begin{theorem}\nConsider an LTI system whose impulse response is $\\bar{c}_{q,k}(n),\\ q>1$, then for a given input $x(n)$ the output $y(n)$ of the system, i.e., $x(n)*\\bar{c}_{q,k}(n)$ is equivalent to computing the first order derivative of $x(n)$.\n\\label{Th1}\n\\end{theorem}\n\\textit{Proof:} Let $x(n) = C$, where $C$ is a constant value, then\n\\begin{equation}\ny(n) = C\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l) = 0,\\ \\text{as }\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l) = 0,\\ \\text{for }q>1.\n\\end{equation}\nIf $x(n) = u(n-n_0)$, where $u(n)$ is an unit step sequence, then \n\\begin{equation}\ny(n) = \\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l)u(n-n_0-l),\n\\end{equation}\nhere $y(n){\\neq}0,\\ \\forall\\ n_0{\\leq}n{\\leq}{n_0+q-2}$. If $x(n) = n$, then\n\\begin{equation}\n\\footnotesize\n\\begin{aligned}\ny(n) &= n\\underbrace{\\sum\\limits_{l=0}^{q-1}\\bar{c}_{q,k}(l)}_{=0}-\\underbrace{\\sum\\limits_{l=0}^{q-1}l\\bar{c}_{q,k}(l)}_{\\mathbf{P}}\\\\\n&= -M\\left[\\sum\\limits_{l=0}^{q-1}le^{\\frac{j2{\\pi}kl}{q}}+\\sum\\limits_{l=0}^{q-1}l e^{\\frac{-j2{\\pi}kl}{q}}\\right]\\\\\n&= M\\left[\\frac{q}{1-e^{\\frac{j2{\\pi}k}{q}}}+\\frac{q}{1-e^{\\frac{-j2{\\pi}k}{q}}}\\right] = Mq.\n\\end{aligned}\n\\label{Der1}\n\\end{equation}\n\\normalsize\nFrom the above analysis, we draw the following conclusions regarding the system output. That is, the system output is:\n\\begin{enumerate}\n\\item Zero for constant input.\n\\item Non-zero at the on transient of the unit step sequence.\n\\item Non-zero constant along the ramps.\n\\end{enumerate}\nIn the \\textit{context of image processing}, any function\/operation satisfying the above three properties is equivalent to first order derivative \\cite{Jain,Gonzalez,7964706}.\nTherefore, $x(n)*\\bar{c}_{q,k}(n)$ is equivalent to computing the first order derivative of $x(n)$.\n\nFurther, with some modifications in the impulse response, the above operation is equivalent to the second order derivative.\nTo be a second order derivative, \nthe system output should satisfy the first two conclusions mentioned in \\textbf{Theorem} \\ref{Th1} and \nit should be zero for $x(n)=n$ \\cite{7964706}.\nThat is, the term $\\mathbf{P}$ in \\eqref{Der1} should be equal to zero, but the assumption of $\\bar{c}_{q,k}(n)$ as impulse response leads to $\\mathbf{P}=Mq$. So, instead of $\\bar{c}_{q,k}(n)$, we try by considering its circular shifts as an impulse response. Therefore,\n\\begin{equation}\n\\footnotesize\n\\sum\\limits_{l=0}^{q-1}l\\bar{c}_{q,k}(l-m) = \\frac{q}{1-cos\\left(v\\right)}\\left[cos\\left(u+v\\right)-cos\\left(u\\right)\\right],\\ 1{\\leq}m{\\leq}q-1,\n\\end{equation}\n\\normalsize\nwhere $u = \\frac{2{\\pi}km}{q}$ and $v = \\frac{2{\\pi}k}{q}$. Now for what value of $m$, \\footnotesize$ cos\\left(u+v\\right)=cos\\left(u\\right)$\\normalsize? Figure \\ref{f1} depicts \\footnotesize$cos\\left(u+v\\right)$ vs $cos\\left(u\\right)\\ $\\normalsize for different $q$ and $k$ values. It is observed that independent of $k$ both the values are equal whenever $q$ is an odd number and $m = \\frac{q-1}{2}$.\n\\begin{figure}[!h]\n\\centering\n \\includegraphics[width=4.3in,height=1.8in]{Images\/secondDerivative.eps} \n\\caption[\\small{(a)-(d) $cos\\left(u+v\\right)$ vs $cos\\left(u\\right)$ for different $q$ and $k$ values}]{\\footnotesize{(a)-(d) $cos\\left(u+v\\right)$ vs $cos\\left(u\\right)$ for different $q$ and $k$ values.}}\n\\label{f1}\n\\end{figure}\nHence, we can summarize the above discussion as:\n\\begin{theorem}\nFor a given odd number $q$, the operation of linear convolution between a given signal $x(n)$ and $\\bar{c}_{q,k}\\left(n-\\frac{q-1}{2}\\right)$ is equivalent to computing the second order derivative of $x(n)$.\n\\end{theorem}\n\\subsection{Application}\nEdge detection of an image is crucial in many applications, where the derivative functions are used \\cite{Jain,Gonzalez}. \nIn this work, we address this problem using CCPSs and compare the results with the results obtained using RSs \\cite{7964706}.\nConsider the Lena image and convert it into two one-dimensional signals, namely $x_1(n)$ and $x_2(n)$, by column-wise appending and row-wise appending respectively.\nNow compute $x_1(n)*\\bar{c}_{5,1}(n),\\ x_2(n)*\\bar{c}_{5,1}(n),\\ x_1(n)*\\bar{c}_{5,2}(n)$ and $x_2(n)*\\bar{c}_{5,2}(n)$, the results are depicted in figure \\ref{f2} (a)-(d) respectively.\nFrom the results, it is clear that we can find the edges using CCPSs.\nNote that, performing convolution on $x_1(n)$ and $x_2(n)$ gives better detection of horizontal (Figure \\ref{f2} (a) and (c)) and vertical (Figure \\ref{f2} (b) and (d)) edges respectively.\nLet $\\hat{U}_q = \\{k_1,k_2,\\dots,k_{\\frac{\\varphi(q)}{2}}\\}$, $q{\\in}\\mathbb{N}$, then we can write the relationship between RSs and CCPSs as\n\\begin{equation}\n\\bar{c}_q(n) = \\bar{c}_{q,k_1}(n)+\\bar{c}_{q,k_2}(n)+\\dots+\\bar{c}_{q,k_{\\frac{\\varphi(q)}{2}}}(n).\n\\end{equation}\nUsing this and linearity property of convolution sum, we can write $x_1(n)*\\bar{c}_5(n) = x_1(n)*\\bar{c}_{5,1}(n)+x_1(n)*\\bar{c}_{5,2}(n)$. Figure \\ref{f2} (e)-(f) validates the same.\nSo, fine edge detection can be achieved using CCPSs over RSs, where RSs are integer-valued sequences and CCPSs are real-valued sequences.\nA similar kind of analysis can be done using CCPSs as the second derivative.\n\\begin{figure}[!h]\n\\centerline{\n \\includegraphics[width=3.3in,height=1.8in]{Images\/edgeDetectionUsingCCPS.PNG} }\n\\caption[\\small{(a)-(d) Applying convolution between Lena image and $\\bar{c}_{q,k}(n)$ in both vertical (column-wise appending) and horizontal (row-wise appending) directions respectively: (a)-(b) With $q=5$ and $k=1$. (c)-(d) With $q=5$ and $k=2$. (e) Adding (a) and (c). (f) Convolving Lena image with $\\bar{c}_{5}(n)$ in a vertical direction.}]{\\footnotesize{(a)-(d) Applying convolution between Lena image and $\\bar{c}_{q,k}(n)$ in both vertical (column-wise appending) and horizontal (row-wise appending) directions respectively: (a)-(b) With $q=5$ and $k=1$. (c)-(d) With $q=5$ and $k=2$. (e) Convolving Lena image with $\\bar{c}_{5}(n)$ in a vertical direction. (f) Adding (a) and (c) then subtract the result from (e).}}\n\\label{f2}\n\\end{figure}\n\n\\section{Properties of CCS}\nConstruct a circulant matrix $\\mathbf{D_{q,k}}{\\in}M_{q}(\\mathbb{R})$ as given below\n\\begin{equation}\n\\mathbf{D_{q,k}} = \\begin{bmatrix}\n\\mathbf{\\bar{c}}^0_\\mathbf{{q,k}} & \\dots & \\mathbf{\\bar{c}}^l_\\mathbf{{q,k}}\\dots & \\mathbf{\\bar{c}}^{q-1}_\\mathbf{{q,k}}\n\\end{bmatrix},\n\\end{equation}\nwhere $\\mathbf{\\bar{c}}^l_\\mathbf{{q,k}}$ indicates $l$ times circular downshift of the sequence $\\mathbf{\\bar{c}_{q,k}}$.\nUsing the \\textit{factorization} property of $\\mathbf{D_{q,k}}$, we can write $\\mathbf{D_{q,k}} = \\mathbf{B_{q,k}}\\mathbf{B_{q,k}^H}$, where\n\\begin{equation}\n\\mathbf{B_{q,k}^H} = \\begin{bmatrix}\nS_{q,{q}-k}(0)&S_{q,{q}-k}(1)&\\dots&S_{q,{q}-k}(q-1) \\\\\nS_{q,k}(0)&S_{q,k}(1)&\\dots&S_{q,k}({q}-1)\\\\\n\\end{bmatrix}_{2{\\times}q}.\n\\label{Factorization}\n\\end{equation}\nFrom \\eqref{Factorization}, the column space of $\\mathbf{D_{q,k}}$ is equal to $v_{q,k}$. Moreover, the first two columns of $\\mathbf{D_{q,k}}$ are linearly independent \\cite{Shah}.\nSo, any signal $\\mathbf{x}{\\in}v_{q,k}$ can be written as,\n\\begin{equation}\n\\left[\\mathbf{x}\\right]_{q{\\times}1} =\\begin{bmatrix}\n\\mathbf{\\bar{c}}^0_\\mathbf{{q,k}}& \\mathbf{\\bar{c}}^1_\\mathbf{{q,k}}\n\\end{bmatrix}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right] = \\left[\\mathbf{F_{q,k}}\\right]_{q{\\times}2}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right]_{2{\\times}1}.\n\\label{ccsBasis}\n\\end{equation}\nLet $q_1{\\in}\\mathbb{N}$, $q_2{\\in}\\mathbb{N}$ and $N = lcm(q_1,q_2)$, then the orthogonality property of CCPSs is \\footnotesize$\\left[\\mathbf{\\hat{c}_{q_1,k_1}}^{l_1}\\right]_{N{\\times}1}^T\\left[\\mathbf{\\hat{c}_{q_2,k_2}}^{l_2}\\right]_{N{\\times}1}= 2NMcos\\left(\\frac{2{\\pi}{k_1}(l_1-l_2)}{q_1}\\right)\\delta(q_1-q_2)\\delta(k_1-k_2)$ \\normalsize\\cite{Shah},\nwhere $\\mathbf{\\hat{c}_{q_i,k_i}}^{l_i}$ is obtained by repeating $\\mathbf{\\bar{c}}^{l_i}_\\mathbf{{q_i,k_i}}$ periodically $\\frac{N}{q_i}$ times.\nUsing this we can conclude the following: The vectors in the basis of $v_{q,k}$ are not orthogonal; The subspaces $v_{q_1,k_1}$ and $v_{q_2,k_2}$ are orthogonal to each other.\nWith this basic introduction, now we discuss a few of CCS properties in the following subsections.\n\n\n\\subsection{shift-invariant Subspace}\nA natural question for a signal belongs to CCS is:\nWhat is the effect if we consider a shifted version of the input signal?\nWe answer this question by proving the following theorem:\n\\begin{theorem}\nCCS is a circular shift-invariant subspace, i.e., if $\\mathbf{x}{\\in}v_{q,k}$, then $\\mathbf{x}^l{\\in}v_{q,k}$, here the shift is interpreted as circular shift $(\\text{modulo }q)$.\n\\end{theorem}\n\\textit{Proof:} From \\eqref{ccsBasis}, circular shifting $\\mathbf{x}$ by an amount $l$ gives \n\\begin{equation}\n\\mathbf{x}^l = \\begin{bmatrix}\n\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}} & \\mathbf{\\bar{c}}^{l+1}_\\mathbf{{q,k}}\n\\end{bmatrix}\\left[\\mathbf{\\hat{\\beta}_{q,k}}\\right].\n\\end{equation}\n$\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}}$ for any $l{\\in}\\mathbb{Z}$ is still a column in $\\mathbf{D_{q,k}}$. \nIt implies both $\\mathbf{\\bar{c}}^{l}_\\mathbf{{q,k}}{\\in}v_{q,k}$ and $\\mathbf{\\bar{c}}^{l+1}_\\mathbf{{q,k}}{\\in}v_{q,k}$.\nThis results in $\\mathbf{x}^l{\\in}v_{q,k}$.\nIn fact, using the row-Vandermonde structure of $\\mathbf{B_{q,k}^H}$ one can prove that any two consecutive columns of $\\mathbf{D_{q,k}}$ act as the basis for $v_{q,k}$,\nhence $\\mathbf{x}^l{\\in}v_{q,k}$.\n\nSince CCS is Shift-invariant, it may be useful in applications like wireless communication \\cite{Forsythe}, subspace tracking:\nwhich play a vital role in video surveillance, source localization in radar and sonar, etc., \\cite{8683025}.\n \n\\subsection{Computing the Projections}\nIn CCPT, a signal $\\mathbf{x}{\\in}M_{N,1}(\\mathbb{C})$ is represented as a linear combination of signals belongs to CCSs \\cite{Shah},\n\\begin{equation}\n\\mathbf{x} = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k=1\\\\(k,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\underbrace{\\mathbf{E_{q_i,k}}\\boldsymbol{\\beta}_\\mathbf{{q_i,k}}}_{\\mathbf{x_{q_i,k}}{\\in}v_{q_i,k}} = \\left[\\mathbf{T_N}\\right]_{N{\\times}N}\\left[\\boldsymbol{\\beta}\\right]_{N{\\times}1}.\n\\label{Synthesis}\n\\end{equation}\nHere, $\\mathbf{T_N}$ is the transformation matrix, $\\boldsymbol{\\beta}$ is the transform coefficient vector,\n$\\mathbf{x_{q_i,k}}$ denotes the projection of $\\mathbf{x}$ onto $v_{q_i,k}$,\n$\\mathbf{[E_{q_i,k}]_{N{\\times}2}}= [\\mathbf{F_{q_i,k}},\\dots,\\mathbf{F_{q_i,k}}]^T$ \nand $\\boldsymbol{\\beta}_\\mathbf{{q_i,k}}{\\in}M_{2,1}(\\mathbb{C})$ is the transform coefficients vector corresponds to $v_{q_i,k}$. \nAs given in \\cite{Shah}, a major application of CCPT is \nestimating the period and frequency information of a signal using $\\boldsymbol{\\beta}$.\nWhile the non-orthogonal basis of $v_{q_i,k}$ makes CCPT a non-orthogonal transform, i.e., it requires computation of $\\mathbf{T_N^{-1}}$ to find $\\boldsymbol{\\beta}$. \nAccording to Strassen's algorithm, computing $\\mathbf{T}^{-1}_\\mathbf{N}$ requires a computational complexity of $\\mathcal{O}(N^{2.81})$ \\cite{Cormen}.\nTo use CCPT in an efficient way, one has to overcome this limitation, for this\nwe compute $\\mathbf{x_{q_i,k}}$ instead of $\\boldsymbol{\\beta}$, which can serve for the same purpose.\nSince $\\left[\\mathbf{E_{q_i,k}}\\right]^H\\left[\\mathbf{E_{q_x,k_y}}\\right] = 0,\\ \\forall\\ i{\\neq}x$ (where $q_i|N \\text{ and } q_x|N$), the projection of $\\mathbf{x}$ onto $v_{q_i,k}$ can be computed as follows:\n\\begin{equation}\n\\mathbf{x_{q_i,k}} = \\underbrace{\\mathbf{E_{q_i,k}}(\\mathbf{E_{q_i,k}^H}\\mathbf{E_{q_i,k}})^{-1}\\mathbf{E_{q_i,k}^H}}_{\\mathbf{P_{q_i,k}}{\\in}M_{N}:\\text{ Projection matrix}}\\mathbf{x}.\n\\label{projMatrix}\n\\end{equation}\nHere $\\mathbf{P_{q_i,k}}$ can be further reduced as given below:\n\\begin{equation}\n\\mathbf{P_{q_i,k}} = \\frac{q_i}{N}\\begin{bmatrix}\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\\\\\n\\vdots&\\ddots&\\vdots\\\\\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\n\\end{bmatrix}, \\text{where}\n\\label{redProj}\n\\end{equation} \n\\begin{equation}\n\\mathbf{[\\hat{P}_{q_i,k}]_{q_i{\\times}q_i}} = \\mathbf{F_{q_i,k}}(\\mathbf{F_{q_i,k}^H}\\mathbf{F_{q_i,k}})^{-1}\\mathbf{F_{q_i,k}^H}.\n\\end{equation}\nLet us divide the input signal $\\mathbf{x}$ into $\\frac{N}{q_i}$ blocks, that is $\\mathbf{x} = \\left[\\mathbf{x^{(1)}},\\mathbf{x^{(2)}},\\dots,\\mathbf{x^{(\\frac{N}{q_i})}}\\right]^T$, where $i^{th}$ block $\\mathbf{x^{(i)}}{\\in}M_{{q_i}{\\times}1}$.\nThen, from \\eqref{projMatrix} and \\eqref{redProj} we can write\n\\begin{equation}\n\\mathbf{x_{q_i,k}} = \\frac{q_i}{N}\\begin{bmatrix}\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\\\\\n\\vdots&\\ddots&\\vdots\\\\\n\\mathbf{\\hat{P}_{q_i,k}}&\\dots&\\mathbf{\\hat{P}_{q_i,k}}\n\\end{bmatrix}\\begin{bmatrix}\n\\mathbf{x^{(1)}}\\\\\n\\vdots\\\\\n\\mathbf{x^{(\\frac{N}{q_i})}}\n\\end{bmatrix} = \\begin{bmatrix}\n\\mathbf{y_{q_i,k}}\\\\\n\\vdots\\\\\n\\mathbf{y_{q_i,k}}\n\\end{bmatrix},\n\\label{Proj}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{y_{q_i,k}} =\\frac{q_i}{N}\\mathbf{\\hat{P}_{q_i,k}}\\sum\\limits_{i=1}^{\\frac{N}{q_i}}\\mathbf{x^{(i)}}.\n\\label{Proj1}\n\\end{equation}\nNow consider a matrix $\\mathbf{\\tilde{P}_{q_i,k}} = \\frac{\\mathbf{D_{q_i,k}}}{q_i}$, then $\\mathbf{\\tilde{P}_{q_i,k}}^2 = \\frac{\\mathbf{B_{q_i,k}}\\mathbf{B}^H_\\mathbf{{q_i,k}}\\mathbf{B_{q_i,k}}\\mathbf{B}^H_\\mathbf{{q_i,k}}}{q_i^2} = \\mathbf{\\tilde{P}_{q_i,k}}$, since $\\mathbf{B}^H_\\mathbf{{q_i,k}}\\mathbf{B_{q_i,k}} = q_i\\mathbf{I}$.\nMoreover, the even symmetry property of CCPSs makes $\\mathbf{\\tilde{P}_{q_i,k}}^H = \\mathbf{\\tilde{P}_{q_i,k}}$.\nSo, we can say that $\\mathbf{\\tilde{P}_{q_i,k}}$ is an orthogonal projection matrix. \nSince $v_{q_i,k}$ is the column space of $\\mathbf{D_{q_i,k}}$, \nwe can write $\\mathbf{\\hat{P}_{q_i,k}} = \\mathbf{\\tilde{P}_{q_i,k}}= \\frac{\\mathbf{D_{q_i,k}}}{q_i}$.\nWith this, \\eqref{Proj1} can be modified as,\n\\begin{equation}\n\\mathbf{y_{q_i,k}} =\\frac{1}{N}\\mathbf{D_{q_i,k}}\\sum\\limits_{i=1}^{\\frac{N}{q_i}}\\mathbf{x^{(i)}} =\\frac{1}{N}\\mathbf{D_{q_i,k}}\\mathbf{\\hat{x}} .\n\\label{Proj2}\n\\end{equation}\nSo, the orthogonal projection $\\mathbf{x_{q_i,k}}$ is computed as follows: First, multiply $\\mathbf{\\hat{x}}$ with the circulant matrix $\\mathbf{D_{q_i,k}}$, then multiply the result with a scale factor $\\frac{1}{N}$ to get $\\mathbf{y_{q_i,k}}$. Now repeat $\\mathbf{y_{q_i,k}}$ periodically $\\frac{N}{q_i}$ times to obtain $\\mathbf{x_{q_i,k}}$.\nThe number of multiplications (computational complexity) involved in computing $\\mathbf{x_{q_i,k}}$ are ${q_i}^2+q_i$.\nFrom \\eqref{Synthesis}, there are total of $\\frac{1}{2}\\sum\\limits_{q_i|N}^{}\\varphi(q_i) = \\frac{N}{2}$ number of CCSs are involved in representing $\\mathbf{x}$. \nSo, the total number of multiplications ($M_{total}$) required for computing projections for all $\\frac{N}{2}$ CCSs are \n\\begin{equation}\nM_{total} = \\begin{cases}\n2+\\frac{1}{2}\\sum\\limits_{\\substack{q_i|N\\\\ q_i{\\geq}3}}^{}\\varphi(q_i)(q_i^2+q_i),\\ &\\text{if }2\\nmid N \\\\\n8+\\frac{1}{2}\\sum\\limits_{\\substack{q_i|N\\\\ q_i{\\geq}3}}^{}\\varphi(q_i)(q_i^2+q_i),\\ &\\text{if }2\\mid N \n\\end{cases}.\n\\end{equation}\n\\begin{table}[h]\n\\centering\n\\caption{C\\scriptsize{OMPARISON OF $M_{total}\\ vs\\ \\floor*{N^{2.81}}$ FOR FEW $N$ VALUES}}\n\\label{tab:Comparison}\n\\begin{adjustbox}{max width=\\textwidth}\n\\scalebox{0.8}{\n\\begin{tabular}{|M{1.1cm}|M{1cm}|M{1.2cm}|M{1.4cm}|M{1.4cm}|M{1.4cm}|} \\hline\nN &\t$3$\t&\t$6$ & $8$&\t$32$\t& $82$\t\\\\\t\\hline\n\\small{$M_{total}$} &\t$14$\t&\t$62$ &\t$172$& $9708$\t& $170568$ \\\\\t\\hline\n\\small{$\\floor*{N^{2.81}}$} & $21$\t&\t$153$ &$344$&\t$16961$\t& $238680$ \\\\ \\hline\n\\end{tabular}}\n\\end{adjustbox}\n\\end{table}\nFor few $N$ values both $M_{total}$ and $\\floor*{N^{2.81}}$ are tabulated in Table \\ref{tab:Comparison}. \nFrom the table, we can conclude that $M_{total} << \\mathcal{O}(N^{2.81})$ and there is an approximate of $40\\%$ reduction in the computational complexity for the values given in the table.\nTherefore, estimating the period and frequency information through\nprojection computation requires less computational complexity over the direct inverse computation method.\nApart from this computational advantage, in some applications, it is required to know the existence of certain periods in an observed signal.\nIn such scenarios, computing projections for those CCSs are sufficient.\n\\subsection{Correlation of Sequences in CCS}\n\\textbf{DFT of CCPS:} For a given $q{\\in}\\mathbb{N}$ and $k{\\in}\\hat{U}_q$,\n\\begin{equation}\nDFT[\\bar{c}_{q,k}(n)] = \\bar{C}_{q,k}(K) = \\begin{cases}\nq,\\ \\text{if }K=k\\ \\text{(or)}\\ q-k,\\\\\n0,\\ \\text{Otherwise}.\n\\end{cases}\n\\end{equation}\nLet $r_{c}(l)$ denotes the circular autocorrelation of $\\bar{c}_{q,k}(l)$, then\n\\begin{equation}\n\\nonumber\nDFT[r_{c}(l)] = R_{c}(K) = \\bar{C}_{q,k}(K)\\bar{C}_{q,k}(K) = q\\bar{C}_{q,k}(K).\n\\end{equation}\nTaking IDFT of above equation leads to $r_{c}(l) = q\\bar{c}_{q,k}(l)$.\nUsing this relation, we prove the following Theorem.\n\\begin{theorem} CCS is closed under circular cross-correlation operation, i.e., if $x(n){\\in}v_{q,k}$ and $y(n){\\in}v_{q,k}$ then the circular cross-correlation $r_{xy}(l)$ also belongs to $v_{q,k}$.\n\\end{theorem}\n\\textit{Proof:} Given $x(n){\\in}v_{q,k}$ and $y(n){\\in}v_{q,k}$ then\n\\begin{equation}\n\\nonumber\n\\footnotesize\n\\begin{aligned}\nr_{xy}(l)& = \\sum\\limits_{n=0}^{q-1}\\underbrace{\\left[\\sum\\limits_{m_1=0}^{1}{\\beta_{m_1}}\\bar{c}_{q,k}(n-m_1)\\right]}_{x(n)}\\underbrace{\\left[\\sum\\limits_{m_2=0}^{1}{\\gamma^*_{m_2}}\\bar{c}_{q,k}(n-l-m_2)\\right]}_{y^*(n-l)}\\\\\n& = \\sum\\limits_{m_1=0}^{1}\\sum\\limits_{m_2=0}^{1}{\\beta_{m_1}}{\\gamma^*_{m_2}}\\left[\\sum\\limits_{n=0}^{q-1}\\bar{c}_{q,k}(n-m_1)\\bar{c}_{q,k}(n-l-m_2)\\right]\\\\\n& = \\sum\\limits_{m_1=0}^{1}\\sum\\limits_{m_2=0}^{1}{\\beta_{m_1}}{\\gamma^*_{m_2}}q\\underbrace{\\bar{c}_{q,k}(l+m_2-m_1)}_{{\\in}v_{q,k}}.\n\\end{aligned}\n\\normalsize\n\\end{equation}\nFrom above, $r_{xy}(l)$ is a weighted linear combination of signals belongs to $v_{q,k}$, hence $r_{xy}(l){\\in}v_{q,k}$.\n\nFurther, the autocorrelation of a $N$-length signal $x(n)$ is\n\\begin{equation}\nr_x(l) = \\sum\\limits_{n=0}^{N-1}x(n)x^*(n-l)\n\\end{equation}\nUsing \\eqref{Synthesis}, the above equation can be modified as\n\\footnotesize\n\\begin{equation}\nr_x(l) = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k_1=1\\\\(k_1,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\sum\\limits_{q_j|N}^{}\\sum\\limits_{\\substack{k_2=1\\\\(k_2,q_j)=1}}^{\\floor*{\\frac{q_j}{2}}}\\underbrace{\\left[\\sum\\limits_{n=0}^{N-1}x_{q_i,k_1}(n)x^*_{q_j,k_2}(n-l)\\right]}_{\\mathbf{Q}}.\n\\label{Corr}\n\\end{equation}\n\\normalsize\nwhere $\\mathbf{Q} = \\begin{cases}\n\\frac{N}{q_i}r_{x_{q_i,k_1}}(l),&\\ \\text{if } q_i=q_j\\ \\&\\ k_1=k_2\\\\\n0,&\\ \\text{if }{k_1}{\\neq}{k_2}\n\\end{cases},\n$\nsince CCS is shift-invariant and both $v_{q_i,k_1}$, $v_{q_j,k_2}$ are orthogonal to each other for $k_1{\\neq}k_2$.\nNow substituting $\\mathbf{Q}$ in \\eqref{Corr} leads to\n\\begin{equation}\n\\footnotesize\n\\frac{1}{N}r_x(l) = \\sum\\limits_{q_i|N}^{}\\sum\\limits_{\\substack{k_1=1\\\\(k_1,q_i)=1}}^{\\floor*{\\frac{q_i}{2}}}\\frac{1}{q_i}r_{x_{q_i,k_1}}(l).\n\\normalsize\n\\end{equation}\nThe above result is summarized as follows:\n\\begin{theorem}\nThe circular autocorrelation of a signal is\nequal to the linear combination of \nits projections (onto CCSs) autocorrelation, with a proper normalization.\n\\end{theorem}\nThis property is useful if the given signal is contaminated with additive noise. In such scenarios projecting $r_x(l)$ onto CCSs gives an accurate period and frequency estimation over projecting $x(n)$.\n\n\n\\section{Conclusion}\nIn this work, we have shown how to use CCPSs as first and second order derivatives. Here, the first order derivative is used to find the edges in an image.\nIt is shown that fine edge detection can be achieved using CCPSs over RSs.\nLater we discussed few properties of CCSs, which may find their applications in signal processing.\nIn particular, the signal information can be estimated with lesser computational complexity using CCS projections.\n\\section*{Acknowledgement}\nThe authors would like to thank Mr. Shiv Nadar, the founder and chairman of HCL and Shiv Nadar Foundation.\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Abstract}\nPhylogenomic analyses are subject to bias from convergence in\nmacromolecular compositions and noise from horizontal gene transfer\n(HGT). Accordingly, compositional convergence leads to contradictory\nresults on the phylogeny of taxa such as the ecologically dominant\nSAR11 group of Alphaproteobacteria, which have extremely streamlined,\nA+T-biased genomes. While careful modeling can reduce bias artifacts\ncaused by convergence, the most consistent and robust phylogenetic\nsignal in genomes may lie distributed among encoded functional\nfeatures that govern macromolecular interactions. Here we develop a\nnovel phyloclassification method based on signatures derived from\nbioinformatically defined tRNA Class-Informative Features (CIFs). tRNA\nCIFs are enriched for features that underlie tRNA-protein\ninteractions. Using a simple tRNA-CIF-based phyloclassifier, we\nobtained results consistent with bias-corrected whole proteome\nphylogenomic studies, rejecting monophyly of SAR11 and affiliating\nmost strains with Rhizobiales with strong statistical support. Yet, as\nexpected by their elevated genomic A+T contents, SAR11 and\nRickettsiales tRNA genes are also similarly and distinctly A+T-rich\nwithin Alphaproteobacteria. Using conventional supermatrix methods on\ntotal tRNA sequence data, we could recover the artifactual result of a\nmonophyletic SAR11 grouping with Rickettsiales. Thus tRNA CIF-based\nphyloclassification is more robust to base content convergence than\nsupermatrix phylogenomics with whole tRNA sequences. Also, given the\nnotoriously promiscuous HGT rates of aminoacyl-tRNA synthetase genes,\ntRNA CIF-based phyloclassification may be at least partly robust to\nHGT of network components. We describe how unique features of the\ntRNA-protein interaction network facilitate mining of traits governing\nmacromolecular interactions from genomic data, and discuss why\ninteraction-governing traits may be especially useful to solve\ndifficult problems in microbial classification and phylogeny.\n\n\n\\section*{Author Summary}\nIn this study, we describe a new way to classify living things using\ninformation from whole genomes. First, for a group of related\norganisms, we bioinformatically predict features by which specific\nclasses of tRNAs are recognized by certain proteins or\ncomplexes. Second, we train an artificial neural network to recognize\nwhich code a new, unknown genome belongs to. We apply our method to\nSAR11, one of the most abundant bacteria in the world's oceans. We\nfind that different strains of SAR11 are more distantly related, both\nto each other and to mitochondria, than previously thought. However,\nwith more traditional treatments of whole tRNA sequence data, we\nobtain different results, best explained as artifacts of base content\nconvergence. Our tRNA features are therefore more robust to genomic\nbase content convergence than the tRNAs in which they are embedded;\nthis is additional evidence of their functional importance. The tRNA\nfeatures we study form a clade-specific and slowly diverging ``feature\nnetwork'' that underlies a universally conserved macromolecular\ninteraction network. We discuss on theoretical grounds why traits\ngoverning macromolecular interactions may be especially well-suited to\nresolve deep relationships in the Tree of Life.\n\n\\section*{Introduction}\n\nWhat parts of genomes are most robust to compositional convergence?\nWhat information is most faithfully inherited vertically? The key\nassumptions of compositional stationarity and consistency in gene\nhistories underpin most current approaches in phylogenomics and are\nfrequently violated (reviewed in {\\it e.g.}\\cite{Gribaldo:2002}). HGT\nis so widespread that the very existence of a ``Tree of Life'' has\nbeen questioned~\\cite{Gogarten:2002us,Bapteste:2009ci}. Better\nunderstanding of ancient phylogenetic relationships requires discovery\nof new universal, slowly-evolving phylogenetic markers that are robust\nto compositional convergence and HGT.\n\nThe controversial phylogeny of {\\it Ca.}Pelagibacter ubique (SAR11) is\na case in point. SAR11 make up between a fifth and a third of the\nbacterial biomass in marine and freshwater\necosystems~\\cite{Morris:2002bn}. Adaptations to extreme environmental\nnutrient limitation may explain why SAR11 have very small cell and\ngenome sizes and small fractions of intergenic\nDNA~\\cite{Giovannoni:2005}. While some recent phylogenomic studies\ndefine a clade among SAR11, the largely endoparasitic Rickettsiales,\nand the alphaproteobacterial ancestor of\nmitochondria~\\cite{Williams:2007p5327,Georgiades:2011gx,Thrash:2011kl},\nothers argue that this placement of SAR11 is an artifact of\nindependent convergence towards increased genomic A+T content, and\nthat SAR11 belongs closer to other free-living Alphaproteobacteria\nsuch as the Rhizobiales and\nRhodobacteraceae~\\cite{Brindefalk:2011ei,RodriguezEzpeleta:2012fw,Viklund:2012jr}.\nMonophyly of SAR11 was also recently\nrejected~\\cite{RodriguezEzpeleta:2012fw}.\n\nNonstationary macromolecular compositions are a known source of bias\nin phylogenomics~\\cite{Foster01062004,Losos14122012}. Widespread variation in\nmacromolecular compositions may be associated with loss of DNA repair\npathways in reduced genomes~\\cite{Dale:2003hc,Viklund:2012jr},\nunveiling an inherent A+T-bias of mutation in\nbacteria~\\cite{Hershberg:2010ig} and elevating genomic A+T\ncontent~\\cite{Moran:2002p5737,Lind:2008cs}. A process such as this has\nlikely altered protein and RNA compositions genome-wide in SAR11, and\nif such effects are accounted for, the placement of SAR11 with\nRickettsiales drops away as an apparent\nartifact~\\cite{RodriguezEzpeleta:2012fw,Viklund:2012jr}. Consistent\nwith this interpretation, SAR11 strain HTTC1062 shares a surprising\nand unique codivergence of \\tRNAaa{His} and histidyl-tRNA synthetase\n(HisRS) with a clade of free-living\nAlphaproteobacteria~\\cite{Wang:2007p1040,Ardell:2010du} that likely\narose only once in bacteria~\\cite{Ardell:2006ko}. This synapomorphy\ncontradicts the placement of SAR11 with Rickettsiales.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=3.2in]{AmrineEtAl_Fig1.pdf\n\\caption{\\figcapone}\n\\label{fig:network}\n\\end{figure*}\n\nThis work was motivated to determine whether the entire system of\ntRNA-protein interactions could be exploited to address phylogeny of\nbacteria, particularly SAR11. The highly conserved tRNA-protein\ninteraction network (Fig.~\\ref{fig:network}) has special advantages\nfor comparative systems biological study from genomic data. First, the\ncomponents and interactions of this network are highly\nconserved. Second, bioinformatic mining of interaction-determining\ntraits from genomic tRNA data is favorable because tRNA structures are\nhighly conserved not just across extant taxa but also across different\nfunctional classes of tRNAs\n(``conformity''~\\cite{WOLFSON01012001}). Yet each functional class of\ntRNA must maintain a hierarchy of increasingly specific interactions\nwith various proteins and other factors\n(``identity''~\\cite{Giege:2008}). The conflicting requirements of\nconformity and identity allow structural comparison and contrast to\npredict class-informative traits of tRNAs from sequence data by\nrelatively simple bioinformatic methods~\\cite{Ardell:2010du}. The\nfeatures that govern tRNA-protein interactions diverge across the\nthree domains of life (reviewed in~\\cite{Giege:1998um}) and also\nwithin the domain of bacteria~\\cite{Ardell:2006ko}.\n\nIn prior work, we developed ``function logos'' to predict, at the\nlevel of individual nucleotides before post-transcriptional\nmodification, what genetically templated information in tRNA gene\nsequences is associated to specific functional identity\nclasses~\\cite{Freyhult:2006dr}. We now call these function-logo-based\npredictions Class-Informative Features (CIFs). A tRNA CIF answers a\nquestion like: ``if a tRNA gene from a group of related genomes\ncarries a specific nucleotide at a specific structural position, how\nmuch informaiton do we gain about that tRNAs specific function?''\nSuch information estimates are corrected for biased sampling of\nfunctional classes and sample size effects~\\cite{Freyhult:2006dr}, and\ntheir statistical significance may be\ncalculated~\\cite{Ardell:2010du}. Although an individual bacterial\ngenome does not present enough data to generate a function logo,\nrelated genome data may be lumped, weakly assuming homogeneity of tRNA\nidentity rules (although heterogeneity generally reduces\nsignal). Function logos recover known tRNA identity elements ({\\it\n i.e.} features that govern the specificity of interactions between\ntRNAs and proteins)~\\cite{Giege:1998um}, and more generally, predict\nfeatures governing interactions with class-specific network partners\nsuch as amidotransferases~\\cite{Bailly:2006cs}. A recent molecular\ndynamics study on a \\tRNAaa{Glu}-GluRS (Glutaminal tRNA-synthetase)\ncomplex identified tRNA functional sites involved in intra- and\ninter-molecular allosteric signalling within GluRS that couples\nsubstrate recognition to reaction catalysis~\\cite{Sethi:2009}. The\npredicted sites are correlated with those from proteobacterial\nfunction logos~\\cite{Freyhult:2007jj}.\n\nIn this work, we show that tRNA CIFs have diverged among\nAlphaproteobacteria in a phylogenetically informative manner. Second,\nas phylogenetic markers, tRNA CIFs are more robust to compositional\nconvergence than the tRNA bodies in which they are embedded. Using our\ntRNA-CIF-based phyloclassification approach, we confirm that SAR11 are\npolyphyletic with the majority of strains clustering with the\nfree-living Alphaproteobacteria. Our results have implications for how\nto best mine genomic data for phylogenetic signals.\n\n\\section*{Results}\n\nWe reannotated Alphaproteobacterial tDNA data from tRNAdb-CE\n2011~\\cite{Abe:2011js} and other prepublication genomic data, and\nsplit them into two groups according to whether or not their source\ngenome contained the uniquely derived synapomorphic traits previously\ndescribed~\\cite{Ardell:2006ko}: a gene for \\tRNAaa{His} containing A73\n(using ``Sprinzl coordinates'',~\\cite{Sprinzl:1998vz}) and lacking\ntemplated $-1G$. We could thereby partition the data into an RRCH\nclade (Rhodobacteraceae, Rhizobiales, Caulobacterales,\nHyphomonadaceae), which present the uniquely derived \\tRNAaa{His}, and\nthe RSR grade (Rhodospirillales, Sphingomonadales, and Rickettsiales,\nexcluding SAR11), which present ``normal'' bacterial \\tRNAaa{His} with\nC73 and genomically templated $-1G$. In all, data from 214\nAlphaproteobacterial genomes represented 11644 predicted tRNA\nsequences (8773 sequences unique within genomes and 3064 total unique\nsequences). Our final dataset contained 147 genomes (8597 tRNAs) for\nthe RRCH clade, 59 genomes (2792 tRNAs) for the RSR grade, and 8\ngenomes (255 tRNAs) of SAR11 strains.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig2.pdf\n\\caption{\\figcaptwo}\n\\label{fig:over}\n\\end{figure*}\n\nThe unique traits of the RRCH \\tRNAaa{His} are perfectly associated to\nsubstitutions of key residues in the motif IIb tRNA-binding loops of\nHisRS involved in tRNA recognition~\\cite{Ardell:2006ko}. Seven of\neight SAR11 strains exhibited the unique \\tRNAaa{His}\/HisRS\ncodivergence traits in common with RRCH genomes. In contrast, strain\nHIMB59 presented ancestral bacterial characters in both \\tRNAaa{His}\nand HisRS (Fig. S1). These results immediately suggest that\nHIMB59 is not monophyletic with the other SAR11 strains, consistent\nwith~\\cite{RodriguezEzpeleta:2012fw}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig3.pdf}\n\\caption{\\figcapthree}\n\\label{fig:binclass}\n\\end{figure*}\n\nWe computed function logos~\\cite{Freyhult:2006dr} of the RRCH clade\nand RSR grade to form the basis of a tRNA-CIF-based binary\nphyloclassifier as shown schematically in Fig.~\\ref{fig:over}. To\nreduce bias, we used a Leave-One-Out Cross-Validation (LOOCV)\napproach. For comparison, we also performed LOOCV phyloclassification\nusing sequence profiles of entire tRNAs, with typical results shown in\nFig.~\\ref{fig:binclass}B. Although the tRNA-CIF-based phyloclassifier\n(Fig.~\\ref{fig:binclass}A) was biased positively by the much larger\nRRCH sample size, it achieved better phylogenetic separation of\ngenomes than the total-tRNA-sequence-based phyloclassifier\n(Fig.~\\ref{fig:binclass}B). The Sphingomonadales and Rhodospirillales\nseparated in scores from the Rickettsiales in both classifiers. Most\nimportantly, the tRNA-CIF-based phyloclassifier placed all eight SAR11\ngenomes closer to the RRCH clade and far away from the Rickettsiales\nwith HIMB59 overlapping the Rhodospirillales, while the\ntotal-tRNA-sequence-based phyloclassifier placed all eight SAR11\ngenomes closer to the Rickettsiales. Fig.~S2 shows the effects of\ndifferent treatments of missing data in the total-tRNA-sequence-based\nclassifier. Method ``zero,'' shown in Fig.~\\ref{fig:binclass}B, is\nmost analogous to the method used to generate\nFig.~\\ref{fig:binclass}A. Method ``skip'' (Fig. S2B) shows that\nSAR11 tRNAs share sequence characters in common with the RSR grade\nthat are not seen in the RRCH clade. Methods ``small'' and ``pseudo''\n(Figs. S2C and S2D) show that SAR11 have sequence traits not\nobserved in either RSR or RRCH.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig4.pdf\n\\caption{\\figcapfour}\n\\label{fig:breakout}\n\\end{figure*}\n\nMany other tRNA classes besides \\tRNAaa{His} contribute to the\ndifferentiated classification of RRCH and RSR genomes by the CIF-based\nbinary classifier (Fig.~\\ref{fig:breakout}). Other tRNA classes are\nalso differentiated between these two groups, including \\tRNAaa{Cys},\n\\tRNAaa{Asp}, \\tRNAaa{Glu}, \\tRNA{Ile}{LAU} (symbolized ``J''),\n\\tRNAaa{Lys}, \\tRNAaa{Tyr}. These results extend the observations of\n\\cite{Wang:2007p1040} who discovered unusual base-pair features of\n\\tRNAaa{Glu} in the RRCH clade. In classes for which the RRCH and RSR\ngroups are well-differentiated, HIMB59 uniquely groups with RSR while\nother strains group with RRCH, while for other tRNA classes, all\nputative SAR11 strains lie outside the RRCH and RSR\ndistributions. This implies that more diverse Alphaproteobacterial\ngenomic data are necessary to completely resolve the phylogenetic\naffiliation of SAR11 strains, but strongly contradict a monophyletic\naffiliation of SAR11 with Rickettsiales. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig5.pdf\n\\caption{\\figcapfive}\n\\label{fig:comp}\n\\end{figure*}\n\nThe increases in genomic A+T contents in SAR11 and Rickettsiales have\nalso driven elevated A+T contents of their tRNA genes\n(Fig.~\\ref{fig:comp}A). Rickettsiales and SAR11 tRNA genes are both\nnotably elevated in both A and T, and share an overall similarity in\ncomposition distinct from other Alphaproteobacteria. Hierarchical\nclustering of Alphaproteobacterial taxa based on tRNA gene base\ncontents closely group SAR11 and Rickettsiales together\n(Fig.~\\ref{fig:comp}B).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5in]{AmrineEtAl_Fig6.pdf\n\\caption{\\figcapsix}\n\\label{fig:unifrac}\n\\end{figure*}\n\nNonstationary tRNA base content --- convergence to greater A+T content\n--- causes all eight SAR11 strains in our dataset to group with\nRickettsiales using phylogenomic approaches based on total tRNA\nsequence evidence. In a ``supermatrix'' phylogenomic approach,\nconcatenating genes for 28 isoacceptor classes from 169 species (2156\ntotal sites) and using the GTR+Gamma model in RAxML, we estimated a\nMaximum Likelihood tree in which all eight putative SAR11 strains\nbranch together with Rickettsiales (Fig.~S3). For this analysis,\nin 31\\% of instances when isoacceptor genes were picked from a genome,\nwe randomly picked one gene from a set of isoacceptor\nparalogs. However, our results did not depend on which paralog we\npicked. Using a distance-based approach with FastTree, we computed a\nconsensus cladogram over 100 replicate alignments each representing\ndifferent randomized picks over paralogs. As the consensus cladogram\nshows (Fig. S4) each replicate distance tree placed all\neight putative SAR11 strains together with Rickettsiales. The recently\nintroduced tRNA-specific FastUniFrac-based method for microbial\nclassification~\\cite{Widmann:2010ea} also places all SAR11 strains\ntogether with Rickettsiales (Fig.~\\ref{fig:unifrac}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6in]{AmrineEtAl_Fig7.pdf\n\\caption{\\figcapsix}\n\\label{fig:multiway}\n\\end{figure*}\n\nHowever, as shown in Fig.~\\ref{fig:multiway}, a multiway classifier\nbased on tRNA CIFs bins all SAR11 strains with the Rhizobiales except\nfor HIMB59, which bins with the Rhodospirillales, consistent with the\nresults of~\\cite{RodriguezEzpeleta:2012fw}. These results use a\nmultilayer perceptron (MLP) classifier implemented in\nWEKA~\\cite{Hall:2009ud} and only seven taxon-specific CIF-based\nsummary scores. The MLP is the simplest non-linear classifier able to\nhandle the interdependent signals in the CIF-based scores for\ntree-like data~\\cite{duda2012pattern}. In a Leave-One-Out\ncross-classification, all other genomes scored consistently with NCBI\nTaxonomy except three placed in Rhodobacteraceae based on 16S\nribosomal RNA evidence: {\\it Stappia aggregata}, {\\it Labrenzia\n alexandrii} and the denitrifying {\\it Pseudovibrio sp.} JE062. None\nof these genomes scored strongly against Rhodobacteraceae except {\\it\n Pseudovibrio}, which scored four times greater against the\nRhizobiales. \n\nTo assess robustness of our results we performed two controls: we\nbootstrapped sites of tRNA data in each genome to be classified, and\nwe filtered away small CIFs with Gorodkin heights $< 0.5$ bits from\nour models, retrained the classifier and bootstrapped sites\nagain. Generally bootstrap support values correspond to original\nclassification probabilities. All SAR11 strains have support values\n$>80\\%$ as Rhizobiales, majority bootstrap values as Rhizobiales\n(HIMB114 at $70\\%$ with Rickettsiales at $15\\%$ and HTCC7211 at $54\\%$\nwith Rickettsiales at $13\\%$), or plurality bootstrap value as\nRickettsiales (HIM5 at $48\\%$ with Rickettsiales at $18\\%$) except\nHIM59 which had a bootstrap support value of $87\\%$ to be in the\nRhodospirillales. Full bootstrap statistics with these model are\nprovided in Table S1.\n\n\\section*{Discussion}\n\nOur results provide strong, albeit unconventional, evidence that most\nSAR11 strains are affiliated with Rhizobiales, while strain HIMB59 is\naffiliated with Rhodospirillales. These results are entirely\nconsistent with comprehensive phylogenomic studies that control for\nnonstationary macromolecular compositions in\nAlphaproteobacteria~\\cite{Brindefalk:2011ei,RodriguezEzpeleta:2012fw,Viklund:2012jr}\nor a site-rate-filtered analysis~\\cite{Gupta:2007}. Our CIF-based\nmethod works even though SAR11 and Rickettsiales tRNAs have converged\nin base content, so that total tRNA sequence-based phylogenomics gives\nopposite results. tRNA CIFs must be at least partly robust to\ncompositional convergence of the tRNA bodies in which they are\nembedded.\n\nIt is well known that aminoacyl-tRNA synthetases (aaRS) are highly\nprone to\nHGT~\\cite{Doolittle:1998bn,Brown:1999ur,Wolf:1999vo,Woese:2000uy,Andam:2011gh}\nincluding in\nAlphaproteobacteria~\\cite{Ardell:2006ko,Dohm:2006ts,Brindefalk:2006jda}. We\nhypothesize that our tRNA-CIF-based phyloclassifiers are also robust\nto HGT of components of the tRNA-protein interaction network,\nconsistent with \\cite{Shiba:1997te}, who argued that a horizontally\ntransferred aaRS is more likely to functionally ameliorate to a\ntRNA-protein network into which it has been transferred rather than\nremodel that network to accomodate itself. HGT of aaRSs may also\nperturb a network so as to cause a distinct pattern of divergence\n(\\cite{Ardell:2006ko} and this work). Wang {\\it et\n al.}~\\cite{Wang:2007p1040} discuss the possibility that RRCH\n\\tRNAaa{His} and HisRS were co-transferred into an ancestral SAR11\ngenome. However, this fails to explain the correlations of many other\ntRNA traits of SAR11 genomes with the RRCH clade reported\nhere. Further study is needed to address the robustness of our method\nto component HGT.\n\nA more distant relationship between most SAR11 strains and\nRickettsiales actually strengthens the genome streamlining\nhypothesis~\\cite{Giovannoni:2005}. If SAR11 were a true branch within\nRickettsiales, it becomes more difficult to claim that genome\nreduction in SAR11 occurred by a selection-driven evolutionary process\ndistinct from the drift-dominated erosion of genomes in the\nRickettsiales~\\cite{Andersson:1998p7319,Moran:2002p5737,Itoh:2002dr}. By\nthe same token, polyphyly of nominal SAR11 strains implies that the\nextensive similarity in genome structure and other traits between\nHIMB59 and SAR11 reported by~\\cite{Grote:2012ge} may have originated\nindependently. Perhaps convergence in some traits is consistent with\nstreamlining, which could also explain trait-sharing between SAR11 and\n{\\it Prochlorococcus}, marine cyanobacteria also argued to have\nundergone streamlining~\\cite{Dufresne:2005gn}. Clear signs of\ndata-limitation in our study should be taken to mean that better\ntaxonomic sampling will improve our results and could ultimately\nresolve more than two origins of SAR11-type genomes among\nAlphaproteobacteria.\n\nWe extracted accurate and robust phylogenetic signals from tRNA gene\nsequences by first integrating within genomes to identify features\nlikely to govern functional interactions with other\nmacromolecules. Unlike small molecule interactions, macromolecular\ninteractions are mediated by genetically determined structural and\ndynamic complementarities. These are intrinsically relative; a large\n{\\it neutral network}~\\cite{Schuster:1994gf} of\ninteraction-determining features should be compatible with the same\ninteraction network. Coevolutionary divergence --- turnover---of\nfeatures that mediate macromolecular interactions, while conserving\nnetwork architecture, has been described in the transcriptional\nnetworks of yeast~\\cite{Kuo:2010fk,Baker:2011fz} and\nworms~\\cite{Barriere:2012dp} and in post-translational modifications\nunderlying protein-protein interactions~\\cite{Krogan:2012cs}. This\nwork demonstrates that divergence of interaction-governing features is\nphylogenetically informative.\n\nIt remains open how such features diverge, with possibilities\nincluding compensatory nearly neutral mutations~\\cite{Hartl:1996dy},\nfluctuating selection~\\cite{He:2011jg}, adaptive\nreversals~\\cite{Bullaughey:2012dg}, and functionalization of\npre-existent variation~\\cite{Haag:2005ty}. Major changes to\ninteraction interfaces may be sufficient to induce genetic isolation\nbetween related lineages, as discussed for the 16S rRNA- and 23S\nrRNA-based standard model of the ``Tree of Life,'' in which many\nimportant and deep branches associate with large, rare macromolecular\nchanges (``signatures'') in ribosome structure and\nfunction~\\cite{Winker:1991jd,Roberts:2008di,Chen:2010et}.\n\nInteraction-mediating features of macromolecules may be systems\nbiology's answer to the phylogeny problem. Perhaps no other traits of\ngenomes are vertically inherited more consistently than those that\nmediate functional interactions with other macromolecules in the same\nlineage. In fact, the structural and dynamic basis of interaction\namong macromolecular components --- essential to their collaborative\nfunction in a system --- may define a lineage better than any of those\ncomponents can themselves, either alone or in ensemble.\n\n\\section*{Materials and Methods}\n\nSupplementary data packages are provided to reproduce all figures from\nraw data and enable third-party classification of alphaproteobacterial\ngenomes.\n\n\\subsection*{Data}\nThe 2011 release of the tRNAdb-CE\ndatabase~\\cite{Abe:2011js} was downloaded on August 24,\n2011. From this master database, we selected Alphaproteobacteria data\nas specified by NCBI Taxonomy data (downloaded September 24, 2010,\n\\cite{Sayers:2010p4006}). Also using NCBI Taxonomy, we further\ntripartitioned Alphaproteobacterial tRNAdb-CE data into those from the\nRRCH clade, the RSR grade (excluding SAR11), and three SAR11\ngenomes, as documented in Supplementary data for figure 2. Five\nadditional SAR11 genomes (for strains HIMB59, HIMB5, HIMB114, IMCC9063\nand HTCC9565) were obtained from J. Cameron Thrash courtesy of the lab\nof S. Giovannoni. We custom annotated tRNA genes in these genomes as\nthe union of predictions from tRNAscan-SE version 1.3.1 (with {\\tt -B}\noption, \\cite{LoweEddy97}) and Aragorn version\n1.2.34~\\cite{Laslett:2004ih}. We classified initiator tRNAs and\n\\tRNA{Ile}{CAU} using TFAM version 1.4~\\cite{Taquist:2007jl} using a\nmodel previously created to do this based on identifications\nin~\\cite{Silva:2006jc} provided as supplementary data. We aligned\ntRNAs with covea version 2.4.4~\\cite{Eddy:1994ul} and the prokaryotic\ntRNA covariance model~\\cite{LoweEddy97}, removed sites with more than\n97\\% gaps with a bioperl-based utility~\\cite{Stajich:2002}, and edited\nthe alignment manually in Seaview 4.1~\\cite{gouy2010seaview} to remove\nCCA tails and remove sequences with unusual secondary structures. We\nmapped sites to Sprinzl coordinates manually~\\cite{Sprinzl:1998vz} and\nverified by spot-checks against tRNAdb~\\cite{juhling2009trnadb}. We\nadded a gap in the -1 position for all sequences and -1G for\n\\tRNAaa{His} in the RSR group~\\cite{Wang:2007p1040}.\n\n\\subsection*{tRNA CIF Estimation and Binary Classifiers}\nOur tRNA-CIF-based binary phyloclassifier with Leave-One-Out\nCross-Validation (LOO CV) is computed directly from function logos,\nestimated from tDNA alignments as described in~\\cite{Freyhult:2006dr}.\nHere, we define a {\\it feature} $f \\in F$ as a nucleotide $n \\in N$ at\na position $l \\in L$ in a structurally aligned tDNA, where $N =\n\\{A,C,G,T\\}$ and $L$ is the set of all Sprinzl\ncoordinates~\\cite{Sprinzl:1998vz}. The set $F$ of all possible\nfeatures is the Cartesian product $F = N \\times L$. A {\\it functional\n class} or {\\it class} of a tDNA is denoted $c \\in \\mathcal{C}$ where\n$\\mathcal{C} = \\{A,C,D,E,F,G,H,I,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y\\}$ is the\nuniverse of functions we here consider, symbolized by IUPAC one-letter\namino acid codes (for aminoacylation classes), $X$ for initiator\ntRNAs, and $J$ for \\tDNA{Ile}{LAU}. A {\\it taxon set of genomes} or\njust {\\it taxon set} $S \\in \\mathcal{P}(G)$ is a set of genomes, where\n$G$ is the set of all genomes, and $\\mathcal{P}(G)$ is the power set\nof $G$. In this work a genome $G$ is represented by the multiset of\ntDNA sequences it contains, denoted $T_G$. The functional information of\nfeatures is computed with a map $h:(F \\times C \\times \\mathcal{P}(G))\n\\longrightarrow \\mathbb{R}_{\\geq 0}$ from the Cartesian product of\nfeatures, classes and taxon sets to non-negative real numbers. For a\nfeature $f \\in F$, class $c \\in \\mathcal{C}$ and taxon set $S \\in\n\\mathcal{P}(G)$, $h(f,c,S)$ is the fraction of functional information\nor ``Gorodkin height''~\\cite{Gorodkin:1997tv}, measured in bits,\nassociated to that feature, class and taxon set. In this work, for a\ngiven taxon set $S$, a function logo $H(S)$ is the tuple:\n\n\\begin{equation}\nH(S) = \\{(\\alpha,\\beta) \\mid \\beta = h(\\alpha, S), \\forall \\alpha \\in (F \\times \\mathcal{C})\\}.\n\\end{equation}\n\nFurthermore the set $I(S) \\subset (F \\times \\mathcal{C})$ of {\\it tRNA\n Class-Informative Features} for taxon set $S$ is defined:\n\n\\begin{equation}\nI(S) = \\{\\alpha \\in (F \\times \\mathcal{C}) \\mid h(\\alpha, S) > 0\\}.\n\\end{equation}\n\nBriefly, a tRNA Class-Informative Feature is a tRNA structural feature\nthat is informative about the functional classes it associates with,\ngiven the context of tRNA structural features that actually co-occur\namong a taxon set of related cells, and corrected for biased sampling\nof classes and finite sampling of\nsequences~\\cite{Freyhult:2006dr}. Let $A$ denote a set of\nAlphaproteobacterial genomes partitioned into three disjoint subsets\n$X$, $Y$ and $Z$ with $X \\cup Y \\cup Z = A$, representing genomes from\nthe RRCH clade, the RSR grade, and the eight nominal {\\it Ca.}\nPelagibacter strains respectively. To execute Leave-One-Out\nCross-Validation of a tRNA CIF-based binary phyloclassifier for a\ngenome $G \\in A$, we compute a score $S_C(G, S_1, S_2)$, averaging\ncontributions from the multiset $T_G$ of tDNAs in $G$ scored against\ntwo function logos $H(S_1)$ and $H(S_2)$ computed respectively from\ntwo disjoint taxon sets $S_1 \\subset A$ and $S_2 \\subset A$, with $G\n\\notin S_1 \\cup S_2$. In this study, those sets are $X \\setminus G$\nand $Y \\setminus G$, denoted $X_G$ and $Y_G$ respectively. Each tDNA\n$t \\in T_G$ presents a set of features $F_t \\subset F$ and has a\nfunctional class $c_t \\in \\mathcal{C}$ associated to it. The score\n$S_C(G,X_G,Y_G)$ is then defined:\n\n\\begin{equation}\nS_C(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |}\\sum_{t \\in T_G}\\sum_{f \\in\n F_t}h(f,c_t,X_G) - h(f,c_t,Y_G).\n\\label{eqn:cif}\n\\end{equation}\n\nAs controls, we implemented four total-tDNA-sequence based binary\nphyloclassifiers to score a genome $G$. All are slight variations in\nwhich a tRNA $t \\in T_G$ of class $c(t)$ contributes a score that is a\ndifference in log relative frequencies of the features it shares in\nclass-specific profile models generated from $X_G$ and $Y_G$. The\ndefault ``zero'' scoring scheme method $S^Z_T(G,X_G, Y_G)$ shown in\nFig.~\\ref{fig:binclass}B is defined as:\n\n\\begin{equation}\nS^Z_T(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in\n F_t}\\log_2 \\frac{p^*(f | c_t, X_G)}{p^*(f | c_t, Y_G)}, \n\\label{weaksol}\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\np^*(f | c, S) \\equiv \n\\begin{cases} \n\\#\\{f , c, S\\} \/ \\#\\{c,S\\} & \\#\\{f , c, S \\} > 0 \\\\ \n1 & \\#\\{f , c, S \\} = 0 \n\\end{cases},\n\\end{eqnarray}\n\n\\noindent $\\#\\{f, c, S \\}$ is the observed frequency of feature $f$ in\ntDNAs of class $c$ in set $S$, and $\\#\\{c, S \\}$ is the frequency of\ntDNAs of class $c$ in set $S$. \n\nMethod ``skip'' corresponds to scoring scheme $S^K_T(G,X_G,\nY_G)$ defined as:\n\n \\begin{equation}\nS^K_T(G,X_G, Y_G) \\equiv \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in F_t} s^k(f,c_t, X_G, Y_G),\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\ns^k(f, c, S, T) \\equiv \n\\begin{cases} \n\\log_2 \\frac{p(f | c, S)}{p(f | c, T)} & \\#\\{f , c, S \\} > 0 \\wedge \\#\\{f , c, T \\} > 0 \\\\\n 0 & \\#\\{f , c, S \\} = 0 \\vee \\#\\{f , c, T \\} = 0 \\end{cases},\n\\end{eqnarray}\n\n\\noindent and $p(f | c, R) \\equiv \\#\\{f , c, R \\}\/ \\#\\{c,R\\}$ for $R\n\\in \\{S,T\\}$ as before.\n\n\n\nMethods ``pseudo'' and ``small'' correspond to scoring schemes $S^I_T(G,X_G,\nY_G)$:\n\n \\begin{equation}\nS^I_T(G,X_G, Y_G) \\equiv \n \\frac{1}{| T_G |} \\sum_{t \\in T_G}\\sum_{f \\in F_t} \\log_2\n \\frac{p^I(f | c_t, X_G)}{p^I(f | c_t, Y_G)},\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\np^I(f | c, S) \\equiv \n\\begin{cases} o \/ t & \\forall n \\in N: \\#\\{(n,l) , c, S \\} > 0 \\\\ \n\\frac{o + I}{t + 4I} & \\exists n \\in N: \\#\\{(n,l) , c, S \\} = 0\n\\end{cases},\n\\end{eqnarray}\n\n\\noindent where $f = (n,l)$, $o \\equiv \\#\\{f, c, S \\}$, $t \\equiv \\#\\{c,S\\}$,\n$I = 1$ for method ``pseudo,'' and, for method ``small,'' $I = 1 \/\nT_A$, where $T_A = \\sum_{G \\in A} T_G$.\n\n\\subsection*{Analysis of tRNA Base Composition}\nWe computed the base composition of tRNAs aggregated by clades using\nbioperl-based~\\cite{Stajich:2002} scripts, and transformed them by the\ncentered log ratio transformation~\\cite{aitchison1986statistical} with\na custom script provided as supplementary data. We then computed\nEuclidean distances on the transformed composition data, and then\nperformed hierarchical clustering by UPGMA on those distances as implemented in\nthe program NEIGHBOR from Phylip 3.6b~\\cite{PHYLIP} and visualized in\nFigTree v.1.4.\n\n\\subsection*{Supermatrix and FastUniFrac Analysis}\nFor supermatrix approaches, we created concatenated tRNA alignments\nfrom 169 Alphaproteobacteria genomes (117 RRCH, 44 RSR, 8 PEL) that\nall shared the same 28 isoacceptors with 77 sites per gene (2156 total\nsites). In cases where a species contained more than a single\nisoacceptor, one was chosen at random. Using a GTR+Gamma model, we\nran RAxML by means of The iPlant\nCollaborative project RAxML server\n(\\url{http:\/\/www.iplantcollaborative.org}, \\cite{Stamatakis01102008})\non January 23, 2013 with their installment of RAxML version\n7.2.8-Alpha (executable raxmlHPC-SSE3, a sequential version of RAxML\noptimized for parallelization). We tested the robustness of our\nresult to random picking of isoacceptors by creating 100 replicate\nconcatenated alignments and running them through\nFastTree~\\cite{Price:2010vy}. For the FastUniFrac\nanalysis we used the FastUniFrac~\\cite{Hamady:2010dr}\nweb-server at \\url{http:\/\/bmf2.colorado.edu\/fastunifrac\/} to\naccomodate our large dataset. We removed two genomes from our dataset\nfor containing fewer than 20 tRNAs, and following~\\cite{Widmann:2010ea}\nremoved anticodon sites. Following~\\cite{Widmann:2010ea}\ndeliberately, we computed an approximate ML tree based on Jukes-Cantor\ndistances using FastTree~\\cite{Price:2010vy}. We then\nqueried the FastUniFrac webserver with this tree, defining\nenvironments as genomes. We then computed a UPGMA tree based on the\nserver's output FastUniFrac distance matrix in NEIGHBOR from Phylip\n3.6b~\\cite{PHYLIP}.\n\n\\subsection*{Multiway Classifier}\nAll tDNA data from the RSR and RRCH clades were partitioned into one\nof seven monophyletic clades: orders Rickettsiales (N = 40 genomes),\nRhodospirillales (N = 10), Sphingomonadales (N = 9), Rhizobiales (N =\n91), and Caulobacterales (N = 6), or families Rhodobacteraceae (N =\n43) or Hyphomonadaceae (N = 4) as specified by NCBI taxonomy\n(downloaded September 24, 2010,~\\cite{Sayers:2010p4006}) and\ndocumented in supplementary data for figure 7. We withheld data from\nthe eight nominal SAR11 strains, as well as from three genera {\\it\n Stappia}, {\\it Pseudovibrio}, and {\\it Labrenzia}, based on\npreliminary analysis of tDNA and CIF sequence variation. Following a\nrelated strategy as with the binary classifier, we computed, for each\ngenome, seven tRNA-CIF-based scores, one for each of the seven\nAlphaproteobacterial clades as represented by their function logos,\nusing the principle of Leave-One-Out Cross-Validation (LOO CV), that\nis, excluding data from the genome to be scored. Function logos were\ncomputed for each clade as described in~\\cite{Freyhult:2006dr}. For\neach taxon set $X_G$ (with genome $G$ left out if it occurs), genome\n$G$ obtains a score $S^M(G,X_G)$ defined by:\n\n\\begin{equation}\nS_M(G,X_G) \\equiv \\frac{1}{| T_G |}\\sum_{t \\in T_G} \\sum_{f \\in\n F_t}h(f,c_t,X_G).\n\\label{eqn:multi}\n\\end{equation}\n\nEach genome $G$ is then represented by a vector of seven scores, one\nfor each taxon set modeled. These labeled vectors were then used to\ntrain a multilayer perceptron classifier in WEKA 3.7.7~(downloaded\nJanuary 24, 2012,~\\cite{Hall:2009ud}) by their defaults through the\ncommand-line interface, which include a ten-fold cross-validation\nprocedure. We bootstrap resampled sites in genomic tRNA alignment data\n(100 replicates) and also bootstrap resampled a reduced (and\nretrained) model including only CIFs with a Gorodkin height~\\cite{Freyhult:2006dr} $\\ge\n0.5$ bits.\n\n\n\\section*{Acknowledgments}\n We thank J. Cameron Thrash and Stephen Giovannoni for sharing data\n in advance of publication, Harish Bhat, Torgeir Hvidsten, Carolin\n Frank and Suzanne Sindi for helpful suggestions.\n\n\n\\bibliographystyle{plos2009}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction and motivation}\n\n\n\n\n\n\nIn this paper, we aim to classify Grothendieck topologies on a poset. This is motivated by the research project of describing quantum theory in terms of topos theory (see e.g., \\cite{BI}, \\cite{DI}, \\cite{HLS}). Here the central object of research are the topoi $\\Sets^{\\CCC(\\A)}$ and $\\Sets^{\\CCC(\\A)^\\op}$, where $\\CCC(\\A)$ is the poset of the commutative C*-subalgebras of some C*-algebra $\\A$ ordered by inclusion. $\\A$ describes some quantum system, whereas the elements of $\\CCC(\\A)$ are interpreted as ``classical snapshots of reality''. The motivation behind this approach is Niels Bohr's doctrine of classical concepts, which, roughly speaking, states that a measurement provides a classical snapshot of quantum reality, and knowledge of all classical snapshots should provide a picture of quantum reality that is as complete as (humanly) possible. In terms of C*-algebras, this means we should be able to reconstruct the algebra $\\A$ in some way from the posetal structure of $\\CCC(\\A)$. Partial results in this direction can be found in \\cite{DH} and \\cite{Hamhalter}, where, at least for certain classes of C*-algebras and von Neumann algebras, it is proven that the Jordan structure of $\\A$ may be recovered from the poset $\\CCC(\\A)$ of commutative subalgebras of $\\A$. Since there is a C*-algebra $\\A$ that is not isomorphic to its opposite C*-algebra $\\A^0$, although $\\CCC(\\A)\\cong\\CCC(\\A^0)$ as posets (see \\cite{Connes}), it is known that the C*-algebraic structure of a C*-algebra cannot always be completely recovered from the posetal structure of $\\CCC(\\A)$. Therefore, extra information is needed, which might be found in sheaf theory, for instance in the form of a Grothendieck topology on $\\CCC(\\A)$.\n\nThe simplest C*-algebras are the finite-dimensional algebras $\\B(\\C^n)$ of $n\\times n$-matrices, where $n$ is a finite positive integer. However, even the posets $\\CCC_n=\\CCC(\\B(\\C^n))$ are already far from trivial (see for instance \\cite[Example 5.3.5]{Heunen}), hence interesting. Familiar with the posetal structure of $\\CCC_n$, one easily sees that these posets are Artinian, i.e., posets for which every non-empty downwards directed subset contains a least element. Moreover, we show in Appendix A that $\\CC(\\A)$ is Artinian if and only if $\\A$ is finite dimensional. The Artinian property is extremely powerful, since it allows one to use a generalization of the principle of induction called \\emph{Artinian induction}\\index{Artinian induction}.\n\nTwo main sources of this paper are \\cite{J&T} and \\cite{EGP}. In the first, several order-theoretic notions are related to the notion of Grothendieck topologies, whilst in the second, the relation is between these order-theoretic notions and the Artinian property is investigated. This article aims to combine both articles. We define a family of Grothendieck topologies on a poset $\\P$ generated by a subset of $\\P$ and show that these Grothendieck topologies in fact exhaust the possible Grothendieck topologies on the given poset if and only $\\P$ is Artinian, where we use Artinian induction for the 'if' direction.\n\nA poset equipped with a Grothendieck topology is called a \\emph{site}. If the extra structure on a $\\CCC(\\A)$ is indeed given by a Grothendieck topology, we also need a notion of 'morphisms of sites'. Finally, we calculate the sheaves corresponding to all classes of Grothendieck topologies we have found. For this calculation, we make use of a posetal version of a sheaf-theoretic result known as the Comparison Lemma, going back to Grothendieck et al \\cite{SGA4} (see also \\cite{Elephant2}), which relates sheaves on a given category to sheaves on some subcategory.\n\n\nIn the appendix, we show that the notion of a Grothendieck topology on $\\P$ is equivalent with the notion of a frame quotient of the frame $\\D(\\P)$ of down-sets of $\\P$, which in turn is equivalent with the notion of a nuclei on $\\D(\\P)$, a congruence on $\\D(\\P)$, a sublocale of $\\D(\\P)$ if we consider $\\D(\\P)$ as a locale. Since all these notions are equivalent, this gives a description of nuclei, congruences, and sublocales of $\\D(\\P)$ that correspond to Grothendieck topologies corresponding to subsets of $\\P$.\n\n\n\n\\section{Preliminaries on order theory}\\label{Order Theory}\\label{Noetherian induction}\nThis section can be skipped if one is familiar with the basic concepts of order theory. We refer to \\cite{DP} for a detailed exposition of order theory.\n\n\\begin{definition}\n A \\emph{poset} $(\\P,\\leq)$ is a set $\\P$ equipped with a \\emph{(partial) order}\\index{order!partial} $\\leq$. That is, $\\leq$, is binary relation, which is reflexive, antisymmetric and transitive. We often write $\\P$ instead of $(\\P,\\leq)$ if it is clear which order is used. If either $p\\leq q$ or $q\\leq p$ for each $p,q\\in\\P$, we say that $\\leq$ is a \\emph{linear order}\\index{order!linear}, and we call $\\P$ a \\emph{linearly ordered set}\\index{linearly ordered set}. Given a poset $\\P$ with order $\\leq$, we define the opposite poset $\\P^\\op$ as the poset with the same underlying set $\\P$, but where $p\\leq q$ if and only if $q\\leq p$ in the original order. .\n\\end{definition}\n\n\n\\begin{definition}\n Let $(\\P,\\leq)$ be a poset and $M\\subseteq \\P$ a subset. We say that $M$ is an \\emph{up-set}\\index{up-set} if for each $x\\in M$ and $y\\in \\P$ we have $x\\leq y$ implies $y\\in M$. Similarly, $M$ is called a \\emph{down-set}\\index{down-set} if for each $x\\in M$ and $y\\in \\P$ we have $x\\geq y$ implies $y\\in M$. Given an element $x\\in \\P$, we define the up-set and down-set generated by $x$ by $\\up x=\\{y\\in \\P:y\\geq x\\}$ and $\\down x=\\{y\\in \\P:y\\leq x\\}$, respectively. We can also define the up-set generated by a subset $M$ of $\\P$ by $\\up M=\\{x\\in \\P:x\\geq m$ for some $m\\in M\\}=\\bigcup_{m\\in M}\\up m$, and similarly, we define the down-set generated by $M$ by $\\down M=\\bigcup_{m\\in M}\\down m$. We denote the collection of all up-sets of a poset $\\P$ by $\\U(\\P)$ and the set of all down-sets by $\\D(\\P)$. If we want to emphasize that we use the order $\\leq$, we write $\\U(\\P,\\leq)$ instead of $\\U(\\P)$.\n\\end{definition}\n\n\\begin{lemma}\\label{cosievegen}\nLet $\\P$ be a poset and $M\\subseteq \\P$. Then $\\down M$ is the smallest down-set containing $M$, and $M=\\down M$ if and only if $M$ is an down-set.\n\\end{lemma}\n\n\n\\begin{definition}\n Let $M$ be a non-empty subset of a poset $\\P$.\n\\begin{enumerate}\n \\item An element $x\\in M$ such that $\\up x\\cap M=\\{x\\}$ is called a \\emph{maximal element}\\index{maximal element}. The set of maximal elements of $M$ is denoted by $\\max M$. The set of all \\emph{minimal elements}\\index{minimal element} $\\min M$ is defined dually.\n \\item If there is an element $x\\in M$ such that $x\\geq y$ for each $y\\in M$, we call $x$ the \\emph{greatest element}\\index{greatest element} of $M$, which is necesarrily unique. The \\emph{least element}\\index{least element} of $M$ is defined dually. If $\\P$ itself has a greatest element, it is denoted by $1$. Dually, the least element of $\\P$ is denoted by $0$.\n \\item An element $x\\in \\P$ is called an \\emph{upper bound}\\index{upper bound} of $M$ if $y\\leq x$ for each $y\\in M$. If the set of upper bounds of $M$ has a least element, it is called the \\emph{join}\\index{join} of $M$, which is necesarrily unique and is denoted by $\\bigvee M$. Dually, we can define \\emph{lower bounds}\\index{lower bound} and the \\emph{meet}\\index{meet} of $M$, which is denoted by $\\bigwedge M$. The (binary) join and the (binary) meet of $\\{x,y\\}$ are denoted by $x\\vee y$ and $x\\wedge y$, respectively.\n \\item $M$ is called upwards (downwards) \\emph{directed}\\index{directed subset of a poset} if for every $x,y\\in M$ there is an upper (lower) bound $z\\in M$.\n\\item $M$ is called a \\emph{filter}\\index{filter of a poset} if $M\\in\\U(\\P)$ and $M$ is downwards directed. We denote the set of filters of $\\P$ by $\\F(\\P)$. A filter is called \\emph{principal}\\index{principal filter} if it is equal to $\\up p$ for some $p\\in \\P$.\n\\item $M$ is called an \\emph{ideal}\\index{ideal of a poset} if $M\\in\\D(\\P)$ and $M$ is upwards directed. We denote the set of ideals of $\\P$ by $\\Idl(\\P)$. An ideal is called \\emph{principal}\\index{principal ideal} if it is equal to $\\down p$ for some $p\\in \\P$.\n\\end{enumerate}\n\\end{definition}\nIf we consider $\\P$ as a category, meets are exactly products, joins are coproducts, and the greatest (least) element of $\\P$ is exactly the terminal (initial) object. These concepts are unique in posets, whereas in arbitrary categories they are only unique up to isomorphism. This follows from the fact that there is at most one morphism between two elements in a poset, hence isomorphic elements in a poset are automatically equal.\n\n\n\\begin{definition}\nLet $\\P$ be a poset. Then $\\P$ is called\n\\begin{enumerate}\n \\item \\emph{Artinian}\\index{Artinian poset} (\\emph{Noetherian})\\index{Noetherian poset} if every non-empty subset contains a minimal (maximal) element;\n \\item a \\emph{meet-semilattice}\\index{meet-semilattice} (\\emph{join-semilattice})\\index{join-semilattice} if all binary meets (joins) exist;\n \\item a \\emph{lattice}\\index{lattice} if all binary meets and binary joins exist;\n \\item a \\emph{complete lattice}\\index{complete lattice} if all meets and joins exist.\n\\item a \\emph{distributive lattice}\\index{distributive lattice} if it is a lattice such that the \\emph{distributive law}\\index{distributive law}\n \\begin{equation}\n x\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z)\n \\end{equation}\n holds for each $x,y,z\\in \\P$\n \\item a \\emph{frame} if all joins and all finite meets exists and the \\emph{infinite distributive law}\\index{infinite distributive law}\n \\begin{equation}\\label{eq:infdist}\n x\\wedge\\bigvee_i y_i=\\bigvee_i(x\\wedge y_i)\n \\end{equation}\n holds for each element $x$ and each family $y_i$ in $\\P$.\n\\end{enumerate}\n\\end{definition}\n\nRemark that in a meet-semilattice (and therefore also in a lattice and in a frame), we have $x\\wedge y=x$ if and only if $x\\leq y$.\n\n\n\\begin{lemma}\\label{lem:completelattice}\n Let $\\P$ be a poset. If $\\P$ has all meets, then $\\P$ becomes a complete lattice with join operation defined by\n\\begin{equation*}\n \\bigvee S=\\bigwedge\\{p\\in\\P:s\\leq p\\ \\forall s\\in S\\}\n\\end{equation*}\nfor each $S\\subseteq\\P$. Dually, if $\\P$ has all joins, then $\\P$ becomes a complete lattice with meet operation defined by\n\\begin{equation*}\n\\bigwedge S=\\bigvee\\{p\\in\\P:p\\leq s\\ \\forall s\\in S\\}\n\\end{equation*}\nfor each $S\\subseteq\\P$. In particular a frame is a complete lattice.\n\\end{lemma}\nAs a consequence, if $\\P$ is a complete lattice, $\\P^\\op$ is a complete lattice as well.\n\n\n\n\nFinally we define morphisms between posets as follows.\n\\begin{definition}\n Let $\\P,\\mathbf{Q}$ be posets and $f:\\P\\to \\mathbf{Q}$ a map. Then $f$ is called\n\\begin{enumerate}\n \\item an \\emph{order morphism} if $x\\leq y$ implies $f(x)\\leq f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{embedding of posets} if $f(x)\\leq f(y)$ if and only if $x\\leq y$ for each $x,y\\in\\P$;\n \\item a \\emph{meet-semilattice morphism} if $\\P$ and $\\mathbf{Q}$ are both meet-semilattices and\\\\ $f(x\\wedge y)=f(x)\\wedge f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{lattice morphism} if $\\P$ and $\\mathbf{Q}$ are both lattices and $f$ is a meet-semilattice morphism such that $f(x\\vee y)=f(x)\\vee f(y)$ for each $x,y\\in \\P$;\n \\item a \\emph{frame morphism} if $\\P$ and $\\mathbf{Q}$ are both frames and $f\\left(\\bigvee M\\right)=\\bigvee f(M)$,\\\\ $f\\left(\\bigwedge N\\right)=\\bigwedge f(N)$ for each $M\\subseteq \\P$ and each finite $N\\subseteq \\P$.\n\\end{enumerate}\nAn order morphism, meet-semilattice morphism, lattice morphism, frame morphism is called an \\emph{order isomorphism}, \\emph{meet-semilattice isomorphism}, \\emph{lattice isormphism}, \\emph{frame isomorphism}\\index{isomorphism}, respectively, if there is a morphism $g:\\mathbf{Q}\\to\\P$ of the same type such that $f\\circ g=1_{\\mathbf{Q}}$ and $g\\circ f=1_\\P$.\n\\end{definition}\nClearly an embedding of posets $f$ is injective. If $f(x)=f(y)$, then $f(x)\\leq f(y)$, so $x\\leq y$, and in a similar way, we find $y\\leq x$, so $x=y$.\nThe converse does not always hold. Consider for instance the poset $\\P=\\{p_1,p_2,p_3\\}$ with $p_1,p_2n$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAssume $\\P$ is Artinian and let $M$ a non-empty downwards directed subset of $\\P$. Then $M$ must have a minimal element $x$. Now, if $y\\in M$, then there must be an $z\\in M$ such that $z\\leq x,y$. Since $x$ is minimal, it follows that $z=x$, so $x\\leq y$, whence $x$ is the least element of $M$.\n\nNow, assume that every non-empty downwards directed subset of $\\P$ has a least element. If $x_1\\geq x_2\\geq x_3\\geq \\ldots$ is a descending chain, then $M=\\{x_i\\}_{i\\in\\N}$ is clearly a directed subset, so it has a least element, say $x_n$. So we must have $x_k=x_n$ for all $k>n$, hence $\\P$ satisfies the descending chain condition.\n\nFinally, we show by contraposition that it follows from the descending chain condition that $\\P$ must be Artinian.\nSo assume that $\\P$ does not satisfy the descending chain\ncondition. Using the Axiom of Dependent Choice, we can construct a sequence $x_1\\geq x_2\\geq \\ldots$ that\ndoes not terminate. The set \\mbox{$M=\\{x_n:n\\in\\N\\}$} is then a non-empty\nsubset of $\\P$ without a minimal element. Thus $\\P$ is not\nArtinian.\n\\end{proof}\n\nSince $\\P$ is Noetherian if and only if $\\P^\\op$ is Artinian, we obtain an equivalent characterization of Noetherian posets.\n\\begin{lemma}\\label{lem:equivalentdefinitionsNoetherian}\nLet $\\P$ be a poset. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item $\\P$ is Noetherian;\n\\item All non-empty upwards directed subsets of $\\P$ have a greatest element.\n\\item $\\P$ satisfies the \\emph{ascending chain condition}\\index{ascending chain condition}. That is, if we have a sequence of\nelements $x_1\\leq x_2\\leq \\ldots$ in $\\P$ (an ascending chain), then the sequence stabilizes. That is, there is an\n$n\\in\\N$ such that $x_k=x_{n}$ for all $k>n$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proposition}[Principle of Artinian induction]\\index{Artinian induction}\\label{prop:ArtinianInduction}\nLet $\\P$ be an Artinian poset and $\\PP$ a property such that:\n\\begin{enumerate}\n\\item Induction basis: $\\PP(x)$ is true for each minimal $x\\in \\P$;\n\\item Induction step: $\\PP(y)$ is true for all $y